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The derivative of ln^3(x) is 3ln²(x)/x

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How to calculate the derivative of ln^3(x)

Note that in this post we will be looking at differentiating ln³(x) which is not the same as differentiating ln(3x). Here is our post dealing with how to differentiate ln(3x)

The chain rule is useful for finding the derivative of a function which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate ln(x) (the answer is l/x)
• We know how to differentiate x³ (the answer is 3x²)

This means the chain rule will allow us to perform the differentiation of the expression ln^3x.

Using the chain rule to find the derivative of ln^3x

Although the expression ln³x contains no parenthesis, we can still view it as a composite function (a function of a function).

We can write ln³x as (ln(x))³.

Now the function is in the form of x³, except it does not have x as the base, instead it has another function of x (ln(x)) as the base.

Let’s call the function of the base g(x), which means:

g(x) = ln(x)

From this it follows that:

(ln(x))³ = g(x)³

So if the function f(x) = x³ and the function g(x) = ln(x), then the function (ln(x))³ can be written as a composite function.

f(x) = x³

f(g(x)) = g(x)³ (but g(x) = ln(x))

f(g(x)) = (ln(x))³

Let’s define this composite function as F(x):

F(x) = f(g(x)) = (ln(x))³

We can find the derivative of ln^3x (F'(x)) by making use of the chain rule.

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The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

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Now we can just plug f(x) and g(x) into the chain rule.

How to find the derivative of ln^3x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(1/x)	g(x) = ln(x) ⇒ g'(x) = 1/x
	= (3ln2(x)).(1/x))	f(g(x)) = (ln(x))3 ⇒ f'(g(x)) = 3ln2(x)
	= 3ln2(x)/x

Using the chain rule, the derivative of ln^3x is 3ln^2(x)/x

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Finally, just a note on syntax and notation: ln^3x is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

ln3x	► Derivative of ln3x = 3ln2(x)/x
ln^3x	► Derivative of ln^3x = 3ln2(x)/x
ln 3 x	► Derivative of ln 3 x = 3ln2(x)/x
(lnx)^3	► Derivative of (lnx)^3 = 3ln2(x)/x
ln cubed x	► Derivative of ln cubed x = 3ln2(x)/x
lnx3	► Derivative of lnx3 = 3ln2(x)/x
ln^3	► Derivative of ln^3 = 3ln2(x)/x

The Second Derivative Of ln^3x

To calculate the second derivative of a function, you just differentiate the first derivative.

From above, we found that the first derivative of ln^3x = 3ln²(x)/x. So to find the second derivative of ln^3x, we just need to differentiate 3ln²(x)/x

We can use a combination of the chain rule and the quotient rule to find the derivative of 3ln²(x)/x.

We can set f(x) = ln²(x) and g(x) = x and apply the quotient rule (and the chain rule on f(x)) to find the derivative of f(x)/g(x) = 3(-ln²(x) + 2ln(x))/x²

► The second derivative of ln^3x is 3(-ln²(x) + 2ln(x))/x²

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The derivative of ln^2(x) is 2ln(x)/x

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How to calculate the derivative of ln^2(x)

Note that in this post we will be looking at differentiating ln²(x) which is not the same as differentiating ln(x²) or ln(2x). Here are our posts dealing with how to differentiate ln(x²) and how to differentiate ln(2x)

There are two methods that can be used for calculating the derivative of ln^2(x).

The first method is by using the product rule for derivatives (since ln²(x) can be written as ln(x).ln(x)).

The second method is by using the chain rule for differentiation.

Finding the derivative of ln^2x using the product rule

The product rule for differentiation states that the derivative of f(x).g(x) is f’(x)g(x) + f(x).g’(x)

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The Product Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(x).g(x)

Then the derivative of F(x) is F'(x) = f’(x)g(x) + f(x)g'(x)

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First, let F(x) = ln²(x)

Then remember that ln²(x) is equal to ln(x).ln(x)

So F(x) = ln(x)ln(x)

By setting f(x) and g(x) as ln(x) means that F(x) = f(x).g(x) and we can apply the product rule to find F'(x) (remembering that the derivative of ln(x) is 1/x)

F'(x)	= f'(x)g(x) + f(x)g'(x)	Product Rule Definition
	= f'(x)ln(x) + ln(x)g'(x)	f(x) = g(x) = ln(x)
	= (1/x)ln(x) + ln(x)(1/x)	f'(x) = g(‘x) = 1/x
	= 2ln(x)/x

Using the product rule, the derivative of ln^2x is 2ln(x)/x

Finding the derivative of ln^2x using the chain rule

The chain rule is useful for finding the derivative of a function which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate ln(x) (the answer is l/x)
• We know how to differentiate x² (the answer is 2x)

This means the chain rule will allow us to perform the differentiation of the expression ln^2x.

Using the chain rule to find the derivative of ln^2x

Although the expression ln²x contains no parenthesis, we can still view it as a composite function (a function of a function).

We can write ln²x as (ln(x))².

Now the function is in the form of x², except it does not have x as the base, instead it has another function of x (ln(x)) as the base.

Let’s call the function of the base g(x), which means:

g(x) = ln(x)

From this it follows that:

(ln(x))² = g(x)²

So if the function f(x) = x² and the function g(x) = ln(x), then the function (ln(x))² can be written as a composite function.

f(x) = x²

f(g(x)) = g(x)² (but g(x) = ln(x))

f(g(x)) = (ln(x))²

Let’s define this composite function as F(x):

F(x) = f(g(x)) = (ln(x))²

We can find the derivative of ln^2x (F'(x)) by making use of the chain rule.

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The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

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Now we can just plug f(x) and g(x) into the chain rule.

How to find the derivative of ln^2x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(1/x)	g(x) = ln(x) ⇒ g'(x) = 1/x
	= (2ln(x)).(1/x))	f(g(x)) = (ln(x))2 ⇒ f'(g(x)) = 2ln(x)
	= 2ln(x)/x

Using the chain rule, the derivative of ln^2x is 2ln(x)/x

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Finally, just a note on syntax and notation: ln^2x is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

ln2x	► Derivative of ln2x = 2ln(x)/x
ln^2x	► Derivative of ln^2x = 2ln(x)/x
ln 2 x	► Derivative of ln 2 x = 2ln(x)/x
(lnx)^2	► Derivative of (lnx)^2 = 2ln(x)/x
ln squared x	► Derivative of ln squared x = 2ln(x)/x
lnx2	► Derivative of lnx2 = 2ln(x)/x
ln^2	► Derivative of ln^2 = 2ln(x)/x

The Second Derivative Of ln^2x

To calculate the second derivative of a function, you just differentiate the first derivative.

From above, we found that the first derivative of ln^2x = 2ln(x)/x. So to find the second derivative of ln^2x, we just need to differentiate 2ln(x)/x

We can use the quotient rule to find the derivative of 2ln(x)/x.

We can set f(x) = ln(x) and g(x) = x and apply the quotient rule to find the derivative of f(x)/g(x) = 2(1-ln(x))/x²

► The second derivative of ln^2x is 2(1-ln(x))/x²

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The derivative of ln(x+1) is 1/(x+1)

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How to calculate the derivative of ln(x+1)

The chain rule is useful for finding the derivative of an expression which could have been differentiated had it been in terms of x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate x+1 (the answer is 1)
• We know how to differentiate ln(x) (the answer is 1/x)

This means the chain rule will allow us to perform the differentiation of the function ln(x+1).

To perform the differentiation, the chain rule says we must differentiate the expression as if it were just in terms of x as long as we then multiply that result by the derivative of what the expression was actually in terms of (in this case the derivative of x+1).

Using the chain rule to find the derivative of ln(x+1)

ln(x+1) is in the form of the standard natural log function ln(x), except it does not have x as an argument, instead it has another function of x (x+1).

Let’s call the function in the argument g(x), which means:

g(x) = x+1

From this it follows that:

ln(x+1) = ln(g(x))

So if the function f(x) = ln(x) and the function g(x) = x+1, then the function ln(x+1) can be written as a composite function.

f(x) = ln(x)

f(g(x)) = ln(g(x)) (but g(x) = x+1)

f(g(x)) = ln(x+1)

Let’s define this composite function as F(x):

F(x) = f(g(x)) = ln(x+1)

We can find the derivative of ln(x+1) (F'(x)) by making use of the chain rule.

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The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

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Now we can just plug f(x) and g(x) into the chain rule. But before we do that, just a recap on the derivative of the natural logarithm.

The derivative of ln(x) with respect to x is (1/x)
The derivative of ln(s) with respect to s is (1/s)

In a similar way, the derivative of ln(x+1) with respect to x+1 is 1/(x+1).
We will use this fact as part of the chain rule to find the derivative of ln(x+1) with respect to x.

How to find the derivative of ln(x+1) using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x)).(1)	g(x) = x+1 ⇒ g'(x) = 1
	= (1/(x+1)).1	f(g(x)) = ln(x+1) ⇒ f'(g(x)) = 1/(x+1) (The derivative of ln(x+1) with respect to x+1 is 1/(x+1)
	= 1/(x+1)

Using the chain rule, we find that the derivative of ln(x+1) is 1/(x+1)

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Finally, just a note on syntax and notation: ln(x+1) is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

lnx+1	► Derivative of lnx+1 =1/(x+1)
ln x+1	► Derivative of ln x+1 = 1/(x+1)
ln x + 1	► Derivative of ln x +1 = 1/(x+1)

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The Second Derivative of ln(x+1)

To calculate the second derivative of a function, you just differentiate the first derivative.

From above, we found that the first derivative of ln(x+1) = 1/(x+1). So to find the second derivative of ln(x+1), we just need to differentiate 1/(x+1).

We can use the quotient rule to find the derivative of 1/(x+1), and we get an answer of -1/(x+1)²

► The second derivative of ln(x+1) = -1/(x+1)²

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The derivative of ln(x^2) is 2/x

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How to calculate the derivative of lnx^2

Note that in this post we will be looking at differentiating ln(x²) which is not the same as differentiating ln²(x) or ln(2x). Here are our posts dealing with how to differentiate ln²(x) and how to differentiate ln(2x)

There are two methods that can be used for calculating the derivative of ln(x²).

The first method is by using the chain rule for derivatives.

The second method is by using the properties of logs to write ln(x²) into a form which differentiable without needing to use the chain rule.

Finding the derivative of ln(x²) using the chain rule

The chain rule is useful for finding the derivative of an expression which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate x²(the answer is 2x)
• We know how to differentiate ln(x) (the answer is 1/x)

This means the chain rule will allow us to perform the differentiation of the function ln(x²).

To perform the differentiation, the chain rule says we must differentiate the expression as if it were just in terms of x as long as we then multiply that result by the derivative of what the expression was actually in terms of (in this case the derivative of x²).

Using the chain rule to find the derivative of ln(x^2)

ln(x²) is in the form of the standard natural log function ln(x), except it does not have x as an argument, instead it has another function of x (x²).

Let’s call the function in the argument g(x), which means:

g(x) = x²

From this it follows that:

ln(x²) = ln(g(x))

So if the function f(x) = ln(x) and the function g(x) = x², then the function ln(x²) can be written as a composite function.

f(x) = ln(x)

f(g(x)) = ln(g(x)) (but g(x) = x²)

f(g(x)) = ln(x²)

Let’s define this composite function as F(x):

F(x) = f(g(x)) = ln(x²)

We can find the derivative of ln(x²) (F'(x)) by making use of the chain rule.

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The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

---

Now we can just plug f(x) and g(x) into the chain rule. But before we do that, just a recap on the derivative of the natural logarithm.

The derivative of ln(x) with respect to x is (1/x)
The derivative of ln(s) with respect to s is (1/s)

In a similar way, the derivative of ln(x²) with respect to x² is (1/x²).
We will use this fact as part of the chain rule to find the derivative of ln(x²) with respect to x.

How to find the derivative of ln(x²) using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x)).(2x)	g(x) = x2⇒ g'(x) = 2x
	= (1/x2).2x	f(g(x)) = ln(x2) ⇒ f'(g(x)) = 1/x2 (The derivative of ln(x2) with respect to x2is (1/x2))
	= 2/x

Using the chain rule, we find that the derivative of ln(x²) is 2/x

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Finally, just a note on syntax and notation: ln(x^2) is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

ln(x2)	► Derivative of ln(x2) = 2/x
lnx2	► Derivative of lnx2 = 2/x
lnx^2	► Derivative of lnx^2 = 2/x
ln x 2	► Derivative of ln x 2 = 2/x
ln x squared	► Derivative of ln x squared = 2/x

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Finding the derivative of ln(x²) using log properties

Since ln is the natural logarithm, the usual properties of logs apply.

The power property of logs states that ln(x^y) = y.ln(x). In other words taking the log of x to a power is the same as multiplying the log of x by that power.

We can therefore use the power rule of logs to rewrite ln(x²) as:

f(x) = ln(x²) = 2.ln(x)

How to find the derivative of ln(x²) using the power rule of logs

f(x)	= 2ln(x)
f'(x)	= 2.(1/x)	The derivative of ln(x) is 1/x
	= 2/x

The Second Derivative of ln(x²)

To calculate the second derivative of a function, you just differentiate the first derivative.

From above, we found that the first derivative of ln(x²) = 2/x. So to find the second derivative of ln(x²), we just need to differentiate 2/x

If we differentiate 2/x we get an answer of (-2/x²).

► The second derivative of ln(x²) = -2/x²

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The derivative of ln(2x^2) is 2/x

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How to calculate the derivative of ln(2x^2)

There are two methods that can be used for calculating the derivative of ln(2x²).

The first method is by using the chain rule for derivatives.

The second method is by using the properties of logs to write ln(2x²) into a form which differentiable without needing to use the chain rule.

Finding the derivative of ln(2x²) using the chain rule

The chain rule is useful for finding the derivative of an expression which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate 2x²(the answer is 4x)
• We know how to differentiate ln(x) (the answer is 1/x)

This means the chain rule will allow us to perform the differentiation of the function ln(2x²).

To perform the differentiation, the chain rule says we must differentiate the expression as if it were just in terms of x as long as we then multiply that result by the derivative of what the expression was actually in terms of (in this case the derivative of 2x²).

Using the chain rule to find the derivative of ln(2x^2)

ln(2x²) is in the form of the standard natural log function ln(x), except it does not have x as an argument, instead it has another function of x (2x²).

Let’s call the function in the argument g(x), which means:

g(x) = 2x²

From this it follows that:

ln(2x²) = ln(g(x))

So if the function f(x) = ln(x) and the function g(x) = 2x², then the function ln(2x²) can be written as a composite function.

f(x) = ln(x)

f(g(x)) = ln(g(x)) (but g(x) = 2x²)

f(g(x)) = ln(2x²)

Let’s define this composite function as F(x):

F(x) = f(g(x)) = ln(2x²)

We can find the derivative of ln(2x²) (F'(x)) by making use of the chain rule.

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The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

---

Now we can just plug f(x) and g(x) into the chain rule. But before we do that, just a recap on the derivative of the natural logarithm.

The derivative of ln(x) with respect to x is (1/x)
The derivative of ln(s) with respect to s is (1/s)

In a similar way, the derivative of ln(2x²) with respect to 2x² is (1/2x²).
We will use this fact as part of the chain rule to find the derivative of ln(2x²) with respect to x.

How to find the derivative of ln(2x²) using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x)).(4x)	g(x) = 2x2⇒ g'(x) = 4x
	= (1/2x2).4x	f(g(x)) = ln(2x2) ⇒ f'(g(x)) = 1/2x2 (The derivative of ln(2x2) with respect to 2x2is (1/2x2))
	= 2/x

Using the chain rule, we find that the derivative of ln(2x²) is 2/x

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Finally, just a note on syntax and notation: ln(2x^2) is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

ln2x2	► Derivative of ln2x2 = 2/x
ln2x^2	► Derivative of ln2x^2 = 2/x
ln 2x 2	► Derivative of ln 2x 2 = 2/x
ln 2 x squared	► Derivative of ln 2 x squared = 2/x

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Finding the derivative of ln(2x²) using log properties

Since ln is the natural logarithm, the usual properties of logs apply.

The product property of logs states that ln(xy) = ln(x) + ln(y). In other words taking the log of a product is equal to the summing the logs of each term of the product.

Since 2x² is the product of 2 and x², we can use the product properties of logs to rewrite ln(2x²):

f(x) = ln(2x²) = ln(2) + ln(x²)

The power property of logs states that ln(x^y) = y.ln(x). In other words taking the log of x to a power is the same as multiplying the log of x by that power.

We can therefore combine the product and power rules of logs to rewrite ln(2x²) as:

f(x) = ln(2x²) = ln(2) + ln(x²) = ln(2) + 2.ln(x)

How to find the derivative of ln(2x²) using the product property of logs

f(x)	= ln(2) + 2ln(x)
f'(x)	= 0 + ln(x)	ln2 is a constant, the derivative of a constant is 0
	= 0 + 2/x	The derivative of ln(x) is 1/x
	= 2/x

The Second Derivative of ln(2x²)

To calculate the second derivative of a function, you just differentiate the first derivative.

From above, we found that the first derivative of ln(2x²) = 2/x. So to find the second derivative of ln(2x²), we just need to differentiate 2/x

If we differentiate 2/x we get an answer of (-2/x²).

► The second derivative of ln(2x²) = -2/x²

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The derivative of ln(2x+1) is 2/(2x+1)

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How to calculate the derivative of ln(2x+1)

The chain rule is useful for finding the derivative of an expression which could have been differentiated had it been in terms of x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate 2x+1 (the answer is 2)
• We know how to differentiate ln(x) (the answer is 1/x)

This means the chain rule will allow us to perform the differentiation of the function ln(2x+1).

To perform the differentiation, the chain rule says we must differentiate the expression as if it were just in terms of x as long as we then multiply that result by the derivative of what the expression was actually in terms of (in this case the derivative of 2x+1).

Using the chain rule to find the derivative of ln(2x+1)

ln(2x+1) is in the form of the standard natural log function ln(x), except it does not have x as an argument, instead it has another function of x (2x+1).

Let’s call the function in the argument g(x), which means:

g(x) = 2x+1

From this it follows that:

ln(2x+1) = ln(g(x))

So if the function f(x) = ln(x) and the function g(x) = 2x+1, then the function ln(2x+1) can be written as a composite function.

f(x) = ln(x)

f(g(x)) = ln(g(x)) (but g(x) = 2x+1)

f(g(x)) = ln(2x+1)

Let’s define this composite function as F(x):

F(x) = f(g(x)) = ln(2x+1)

We can find the derivative of ln(2x+1) (F'(x)) by making use of the chain rule.

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The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

---

Now we can just plug f(x) and g(x) into the chain rule. But before we do that, just a recap on the derivative of the natural logarithm.

The derivative of ln(x) with respect to x is (1/x)
The derivative of ln(s) with respect to s is (1/s)

In a similar way, the derivative of ln(2x+1) with respect to 2x+1 is 1/(2x+1).
We will use this fact as part of the chain rule to find the derivative of ln(2x+1) with respect to x.

How to find the derivative of ln(2x+1) using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x)).(2)	g(x) = 2x+1 ⇒ g'(x) = 2
	= (1/(2x+1)).2	f(g(x)) = ln(2x+1) ⇒ f'(g(x)) = 1/(2x+1) (The derivative of ln(2x+1) with respect to 2x+1 is 1/(2x+1)
	= 2/(2x+1)

Using the chain rule, we find that the derivative of ln(2x+1) is 2/(2x+1)

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Finally, just a note on syntax and notation: ln(2x+1) is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

ln2x+1	► Derivative of ln2x+1 =2/(2x+1)
ln 2x+1	► Derivative of ln 2x+1 = 2/(2x+1)
ln 2 x + 1	► Derivative of ln 2 x +1 = 2/(2x+1)

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The Second Derivative of ln(2x+1)

To calculate the second derivative of a function, you just differentiate the first derivative.

From above, we found that the first derivative of ln(2x+1) = 2/(2x+1). So to find the second derivative of ln(2x+1), we just need to differentiate 2/(2x+1).

We can use the quotient rule to find the derivative of 2/(2x+1), and we get an answer of -4/(2x+1)²

► The second derivative of ln(2x+1) = -4/(2x+1)²

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The derivative of ln(8x) is 1/x

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How to calculate the derivative of ln(8x)

There are two methods that can be used for calculating the derivative of ln(8x).

The first method is by using the chain rule for derivatives.

The second method is by using the properties of logs to write ln(8x) into a form which differentiable without needing to use the chain rule.

Finding the derivative of ln(8x) using the chain rule

The chain rule is useful for finding the derivative of an expression which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate 8x (the answer is 8)
• We know how to differentiate ln(x) (the answer is 1/x)

This means the chain rule will allow us to perform the differentiation of the function ln(8x).

To perform the differentiation, the chain rule says we must differentiate the expression as if it were just in terms of x as long as we then multiply that result by the derivative of what the expression was actually in terms of (in this case the derivative of 8x).

Using the chain rule to find the derivative of ln(8x)

ln(8x) is in the form of the standard natural log function ln(x), except it does not have x as an argument, instead it has another function of x (8x).

Let’s call the function in the argument g(x), which means:

g(x) = 8x

From this it follows that:

ln(8x) = ln(g(x))

So if the function f(x) = ln(x) and the function g(x) = 8x, then the function ln(8x) can be written as a composite function.

f(x) = ln(x)

f(g(x)) = ln(g(x)) (but g(x) = 8x)

f(g(x)) = ln(8x)

Let’s define this composite function as F(x):

F(x) = f(g(x)) = ln(8x)

We can find the derivative of ln(8x) (F'(x)) by making use of the chain rule.

---

The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

---

Now we can just plug f(x) and g(x) into the chain rule.

Now we can just plug f(x) and g(x) into the chain rule. But before we do that, just a quick recap on the derivative of the natural logarithm.

The derivative of ln(x) with respect to x is (1/x)
The derivative of ln(s) with respect to s is (1/s)

In a similar way, the derivative of ln(8x) with respect to 8x is (1/8x).
We will use this fact as part of the chain rule to find the derivative of ln(8x) with respect to x.

How to find the derivative of ln(8x) using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x)).(8)	g(x) = 8x ⇒ g'(x) = 8
	= (1/8x).8	f(g(x)) = ln(8x) ⇒ f'(g(x)) = 1/8x (The derivative of ln(8x) with respect to 8x is (1/8x))
	= 1/x

Using the chain rule, we find that the derivative of ln(8x) is 1/x

---

Finally, just a note on syntax and notation: ln(8x) is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

ln8x	► Derivative of ln8x =1/x
ln 8x	► Derivative of ln 8x = 1/x
ln 8 x	► Derivative of ln 8 x = 1/x

---

Top Tip

It’s possible to generalize the derivative of expressions in the form ln(ax) (where a is a constant value):

The derivative of ln(ax) = 1/x

(Regardless of the value of the constant, the derivative of ln(ax) is always 1/x)

---

Finding the derivative of ln(8x) using log properties

Since ln is the natural logarithm, the usual properties of logs apply.

The product property of logs states that ln(xy) = ln(x) + ln(y). In other words taking the log of a product is equal to the summing the logs of each term of the product.

Since 8x is the product of 8 and x, we can use the product properties of logs to rewrite ln(8x):

f(x) = ln(8x) = ln(8) + ln(x)

How to find the derivative of ln(8x) using the product property of logs

f(x)	= ln(8) + ln(x)
f'(x)	= 0 + ln(x)	ln8 is a constant, the derivative of a constant is 0
	= 0 + 1/x	The derivative of ln(x) is 1/x
	= 1/x

The Second Derivative of ln(8x)

To calculate the second derivative of a function, you just differentiate the first derivative.

From above, we found that the first derivative of ln(8x) = 1/x. So to find the second derivative of ln(8x), we just need to differentiate 1/x

If we differentiate 1/x we get an answer of (-1/x²).

► The second derivative of ln(8x) = -1/x²

---

The derivative of ln(7x) is 1/x

---

How to calculate the derivative of ln(7x)

There are two methods that can be used for calculating the derivative of ln(7x).

The first method is by using the chain rule for derivatives.

The second method is by using the properties of logs to write ln(7x) into a form which differentiable without needing to use the chain rule.

Finding the derivative of ln(7x) using the chain rule

The chain rule is useful for finding the derivative of an expression which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate 7x (the answer is 7)
• We know how to differentiate ln(x) (the answer is 1/x)

This means the chain rule will allow us to perform the differentiation of the function ln(7x).

To perform the differentiation, the chain rule says we must differentiate the expression as if it were just in terms of x as long as we then multiply that result by the derivative of what the expression was actually in terms of (in this case the derivative of 7x).

Using the chain rule to find the derivative of ln(7x)

ln(7x) is in the form of the standard natural log function ln(x), except it does not have x as an argument, instead it has another function of x (7x).

Let’s call the function in the argument g(x), which means:

g(x) = 7x

From this it follows that:

ln(7x) = ln(g(x))

So if the function f(x) = ln(x) and the function g(x) = 7x, then the function ln(7x) can be written as a composite function.

f(x) = ln(x)

f(g(x)) = ln(g(x)) (but g(x) = 7x)

f(g(x)) = ln(7x)

Let’s define this composite function as F(x):

F(x) = f(g(x)) = ln(7x)

We can find the derivative of ln(7x) (F'(x)) by making use of the chain rule.

---

The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

---

Now we can just plug f(x) and g(x) into the chain rule.

Now we can just plug f(x) and g(x) into the chain rule. But before we do that, just a quick recap on the derivative of the natural logarithm.

The derivative of ln(x) with respect to x is (1/x)
The derivative of ln(s) with respect to s is (1/s)

In a similar way, the derivative of ln(7x) with respect to 7x is (1/7x).
We will use this fact as part of the chain rule to find the derivative of ln(7x) with respect to x.

How to find the derivative of ln(7x) using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x)).(7)	g(x) = 7x ⇒ g'(x) = 7
	= (1/7x).7	f(g(x)) = ln(7x) ⇒ f'(g(x)) = 1/7x (The derivative of ln(7x) with respect to 7x is (1/7x))
	= 1/x

Using the chain rule, we find that the derivative of ln(7x) is 1/x

---

Finally, just a note on syntax and notation: ln(7x) is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

ln7x	► Derivative of ln7x =1/x
ln 7x	► Derivative of ln 7x = 1/x
ln 7 x	► Derivative of ln 7 x = 1/x

---

Top Tip

It’s possible to generalize the derivative of expressions in the form ln(ax) (where a is a constant value):

The derivative of ln(ax) = 1/x

(Regardless of the value of the constant, the derivative of ln(ax) is always 1/x)

---

Finding the derivative of ln(7x) using log properties

Since ln is the natural logarithm, the usual properties of logs apply.

The product property of logs states that ln(xy) = ln(x) + ln(y). In other words taking the log of a product is equal to the summing the logs of each term of the product.

Since 7x is the product of 7 and x, we can use the product properties of logs to rewrite ln(7x):

f(x) = ln(7x) = ln(7) + ln(x)

How to find the derivative of ln(7x) using the product property of logs

f(x)	= ln(7) + ln(x)
f'(x)	= 0 + ln(x)	ln7 is a constant, the derivative of a constant is 0
	= 0 + 1/x	The derivative of ln(x) is 1/x
	= 1/x

The Second Derivative of ln(7x)

To calculate the second derivative of a function, you just differentiate the first derivative.

From above, we found that the first derivative of ln(7x) = 1/x. So to find the second derivative of ln(7x), we just need to differentiate 1/x

If we differentiate 1/x we get an answer of (-1/x²).

► The second derivative of ln(7x) = -1/x²

---

The derivative of ln(6x) is 1/x

---

How to calculate the derivative of ln(6x)

There are two methods that can be used for calculating the derivative of ln(6x).

The first method is by using the chain rule for derivatives.

The second method is by using the properties of logs to write ln(6x) into a form which differentiable without needing to use the chain rule.

Finding the derivative of ln(6x) using the chain rule

The chain rule is useful for finding the derivative of an expression which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate 6x (the answer is 6)
• We know how to differentiate ln(x) (the answer is 1/x)

This means the chain rule will allow us to perform the differentiation of the function ln(6x).

To perform the differentiation, the chain rule says we must differentiate the expression as if it were just in terms of x as long as we then multiply that result by the derivative of what the expression was actually in terms of (in this case the derivative of 6x).

Using the chain rule to find the derivative of ln(6x)

ln(6x) is in the form of the standard natural log function ln(x), except it does not have x as an argument, instead it has another function of x (6x).

Let’s call the function in the argument g(x), which means:

g(x) = 6x

From this it follows that:

ln(6x) = ln(g(x))

So if the function f(x) = ln(x) and the function g(x) = 6x, then the function ln(6x) can be written as a composite function.

f(x) = ln(x)

f(g(x)) = ln(g(x)) (but g(x) = 6x)

f(g(x)) = ln(6x)

Let’s define this composite function as F(x):

F(x) = f(g(x)) = ln(6x)

We can find the derivative of ln(6x) (F'(x)) by making use of the chain rule.

---

The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

---

Now we can just plug f(x) and g(x) into the chain rule.

Now we can just plug f(x) and g(x) into the chain rule. But before we do that, just a quick recap on the derivative of the natural logarithm.

The derivative of ln(x) with respect to x is (1/x)
The derivative of ln(s) with respect to s is (1/s)

In a similar way, the derivative of ln(6x) with respect to 6x is (1/6x).
We will use this fact as part of the chain rule to find the derivative of ln(6x) with respect to x.

How to find the derivative of ln(6x) using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x)).(6)	g(x) = 6x ⇒ g'(x) = 6
	= (1/6x).6	f(g(x)) = ln(6x) ⇒ f'(g(x)) = 1/6x (The derivative of ln(6x) with respect to 6x is (1/6x))
	= 1/x

Using the chain rule, we find that the derivative of ln(6x) is 1/x

---

Finally, just a note on syntax and notation: ln(6x) is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

ln6x	► Derivative of ln6x =1/x
ln 6x	► Derivative of ln 6x = 1/x
ln 6 x	► Derivative of ln 6 x = 1/x

---

Top Tip

It’s possible to generalize the derivative of expressions in the form ln(ax) (where a is a constant value):

The derivative of ln(ax) = 1/x

(Regardless of the value of the constant, the derivative of ln(ax) is always 1/x)

---

Finding the derivative of ln(6x) using log properties

Since ln is the natural logarithm, the usual properties of logs apply.

The product property of logs states that ln(xy) = ln(x) + ln(y). In other words taking the log of a product is equal to the summing the logs of each term of the product.

Since 6x is the product of 6 and x, we can use the product properties of logs to rewrite ln(6x):

f(x) = ln(6x) = ln(6) + ln(x)

How to find the derivative of ln(6x) using the product property of logs

f(x)	= ln(6) + ln(x)
f'(x)	= 0 + ln(x)	ln6 is a constant, the derivative of a constant is 0
	= 0 + 1/x	The derivative of ln(x) is 1/x
	= 1/x

The Second Derivative of ln(6x)

To calculate the second derivative of a function, you just differentiate the first derivative.

From above, we found that the first derivative of ln(6x) = 1/x. So to find the second derivative of ln(6x), we just need to differentiate 1/x

If we differentiate 1/x we get an answer of (-1/x²).

► The second derivative of ln(6x) = -1/x²

---

The derivative of ln(5x) is 1/x

---

How to calculate the derivative of ln(5x)

There are two methods that can be used for calculating the derivative of ln(5x).

The first method is by using the chain rule for derivatives.

The second method is by using the properties of logs to write ln(5x) into a form which differentiable without needing to use the chain rule.

Finding the derivative of ln(5x) using the chain rule

The chain rule is useful for finding the derivative of an expression which could have been differentiated had it been in x, but it is in the form of another expression which could also be differentiated if it stood on its own.

In this case:

• We know how to differentiate 5x (the answer is 5)
• We know how to differentiate ln(x) (the answer is 1/x)

This means the chain rule will allow us to perform the differentiation of the function ln(5x).

To perform the differentiation, the chain rule says we must differentiate the expression as if it were just in terms of x as long as we then multiply that result by the derivative of what the expression was actually in terms of (in this case the derivative of 5x).

Using the chain rule to find the derivative of ln(5x)

ln(5x) is in the form of the standard natural log function ln(x), except it does not have x as an argument, instead it has another function of x (5x).

Let’s call the function in the argument g(x), which means:

g(x) = 5x

From this it follows that:

ln(5x) = ln(g(x))

So if the function f(x) = ln(x) and the function g(x) = 5x, then the function ln(5x) can be written as a composite function.

f(x) = ln(x)

f(g(x)) = ln(g(x)) (but g(x) = 5x)

f(g(x)) = ln(5x)

Let’s define this composite function as F(x):

F(x) = f(g(x)) = ln(5x)

We can find the derivative of ln(5x) (F'(x)) by making use of the chain rule.

---

The Chain Rule:
For two differentiable functions f(x) and g(x)

If F(x) = f(g(x))

Then the derivative of F(x) is F'(x) = f’(g(x)).g’(x)

---

Now we can just plug f(x) and g(x) into the chain rule.

Now we can just plug f(x) and g(x) into the chain rule. But before we do that, just a quick recap on the derivative of the natural logarithm.

The derivative of ln(x) with respect to x is (1/x)
The derivative of ln(s) with respect to s is (1/s)

In a similar way, the derivative of ln(5x) with respect to 5x is (1/5x).
We will use this fact as part of the chain rule to find the derivative of ln(5x) with respect to x.

How to find the derivative of ln(5x) using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x)).(5)	g(x) = 5x ⇒ g'(x) = 5
	= (1/5x).5	f(g(x)) = ln(5x) ⇒ f'(g(x)) = 1/5x (The derivative of ln(5x) with respect to 5x is (1/5x))
	= 1/x

Using the chain rule, we find that the derivative of ln(5x) is 1/x

---

Finally, just a note on syntax and notation: ln(5x) is sometimes written in the forms below (with the derivative as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

ln5x	► Derivative of ln5x =1/x
ln 5x	► Derivative of ln 5x = 1/x
ln 5 x	► Derivative of ln 5 x = 1/x

---

Top Tip

It’s possible to generalize the derivative of expressions in the form ln(ax) (where a is a constant value):

The derivative of ln(ax) = 1/x

(Regardless of the value of the constant, the derivative of ln(ax) is always 1/x)

---

Finding the derivative of ln(5x) using log properties

Since ln is the natural logarithm, the usual properties of logs apply.

The product property of logs states that ln(xy) = ln(x) + ln(y). In other words taking the log of a product is equal to the summing the logs of each term of the product.

Since 5x is the product of 5 and x, we can use the product properties of logs to rewrite ln(5x):

f(x) = ln(5x) = ln(5) + ln(x)

How to find the derivative of ln(5x) using the product property of logs

f(x)	= ln(5) + ln(x)
f'(x)	= 0 + ln(x)	ln5 is a constant, the derivative of a constant is 0
	= 0 + 1/x	The derivative of ln(x) is 1/x
	= 1/x

The Second Derivative of ln(5x)

To calculate the second derivative of a function, you just differentiate the first derivative.

From above, we found that the first derivative of ln(5x) = 1/x. So to find the second derivative of ln(5x), we just need to differentiate 1/x

If we differentiate 1/x we get an answer of (-1/x²).

► The second derivative of ln(5x) = -1/x²