The Derivative of ln(3x)

September 28, 2020 ~ DerivativeIt ~ Leave a comment

The derivative of ln(3x) is 1/x

How to calculate the derivative of ln(3x)

There are two methods that can be used for calculating the derivative of ln(3x).

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

The second method is by using the properties of logs to write ln(3x) into a form which differentiable without needing to use the chain rule.

Finding the derivative of ln(3x) 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 3x (the answer is 3)
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(3x).

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 3x).

Using the chain rule to find the derivative of ln(3x)

ln(3x) 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 (3x).

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

g(x) = 3x

From this it follows that:

ln(3x) = ln(g(x))

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

f(x) = ln(x)

f(g(x)) = ln(g(x)) (but g(x) = 3x)

f(g(x)) = ln(3x)

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

F(x) = f(g(x)) = ln(3x)

We can find the derivative of ln(3x) (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(3x) with respect to 3x is (1/3x).
We will use this fact as part of the chain rule to find the derivative of ln(3x) with respect to x.

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)).(3)	g(x) = 3x ⇒ g'(x) = 3
	= (1/3x).3	f(g(x)) = ln(3x) ⇒ f'(g(x)) = 1/3x (The derivative of ln(3x) with respect to 3x is (1/3x))
	= 1/x

Using the chain rule, we find that the derivative of ln(3x) is 1/x

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 =1/x
ln 3x	► Derivative of ln 3x = 1/x
ln 3 x	► Derivative of ln 3 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(3x) 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 3x is the product of 3 and x, we can use the product properties of logs to rewrite ln(3x):

f(x) = ln(3x) = ln(3) + ln(x)

How to find the derivative of ln(3x) using the product property of logs

f(x)	= ln(3) + ln(x)
f'(x)	= 0 + ln(x)	ln3 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(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) = 1/x. So to find the second derivative of ln(3x), we just need to differentiate 1/x

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

► The second derivative of ln(3x) = -1/x²

The Derivative of cos^2x

September 25, 2020 ~ DerivativeIt

The derivative of cos^2x is -sin(2x)

How to calculate the derivative of cos^2x

Note that in this post we will be looking at differentiating cos²(x) which is not the same as differentiating cos(2x). Here is our post dealing with how to differentiate cos(2x).

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

The first method is by using the product rule for derivatives (since cos²(x) can be written as cos(x).cos(x)).

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

Finding the derivative of cos^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)

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)

First, let F(x) = cos²(x)

Then remember that cos²(x) is equal to cos(x).cos(x)

So F(x) = cos(x)cos(x)

By setting f(x) and g(x) as cos(x) means that F(x) = f(x).g(x) and we can apply the product rule to find F'(x)

F'(x)	= f'(x)g(x) + f(x)g'(x)	Product Rule Definition
	= f'(x)cos(x) + cos(x)g'(x)	f(x) = g(x) = cos(x)
	= -sin(x)cos(x) + cos(x)(-sin(x))	f'(x) = g(‘x) = -sin(x)
	= -2sin(x)cos(x)
	= –sin(2x)	Double angle identity: sin(2x) = 2sin(x)cos(x)

Using the product rule, the derivative of cos^2x is -sin(2x)

Finding the derivative of cos^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 cos(x) (the answer is -sin(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 cos^2x.

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

Although the expression cos²x contains no parenthesis, we can still view it as a composite function (a function of a function).

We can write cos²x as (cos(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 (cos(x)) as the base.

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

g(x) = cos(x)

From this it follows that:

cos(x)² = g(x)²

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

f(x) = x²

f(g(x)) = g(x)² (but g(x) = cos(x))

f(g(x)) = (cos(x))²

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

F(x) = f(g(x)) = (cos(x))²

We can find the derivative of cos^2x (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.

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

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(-sin(x))	g(x) = cos(x) ⇒ g'(x) = sin(x)
	= (2cos(x)).(-sin(x))	f(g(x)) = (cos(x))² ⇒ f'(g(x)) = 2cos(x)
	= -2cos(x)sin(x)
	= -sin(2x)	Double angle identity: sin(2x) = 2sin(x)cos(x)

Using the chain rule, the derivative of cos^2x is -sin(2x)

Finally, just a note on syntax and notation: cos^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.

cos²x	► Derivative of cos²x = -sin(2x)
cos^2(x)	► Derivative of cos^2(x) = -sin(2x)
cos 2 x	► Derivative of cos 2 x = -sin(2x)
(cosx)^2	► Derivative of (cosx)^2 = -sin(2x)
cos squared x	► Derivative of cos squared x = -sin(2x)
cosx2	► Derivative of cosx2 = -sin(2x)
cos^2	► Derivative of cos^2 = -sin(2x)

The Second Derivative Of cos^2x

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

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

We can use the chain rule to find the derivative of -sin(2x).

We can set g(x) = 2x and f(x) = -sin(x). the F(x) = f(g(x)) = -sin(2x). We can then apply the chain rule to find F'(x).

The chain rules says that the derivative of F(x) is equal to f’(g(x)).g'(x).

f'(g(x)).g'(x)
= -cos(2x).2
= -2cos(2x)

► The second derivative of cos^2x is -2cos(2x)

The Derivative of e^4x

September 25, 2020 ~ DerivativeIt

The derivative of e^4x is 4e^4x

How to calculate the derivative of e^4x

In this case:

We know how to differentiate e^x (the answer is e^x)
We know how to differentiate 4x (the answer is 4)

Because e^4x is a function which is a combination of e^x and 4x, it means we can perform the differentiation of e to the 4x by making use of the chain rule.

Using the chain rule to find the derivative of e^4x

Although the function e^4x contains no parenthesis, we can still view it as a composite function (a function of a function).

If we add parenthesis around the exponent, we get e^(4x).

Now the function is in the form of the standard exponential function e^x, except it does not have x as an exponent, instead the exponent is another function of x (4x).

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

g(x) = 4x

From this it follows that:

e^4x = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = 4x, the function e^4x can be written as a composite function of f(x) and g(x).

f(x) = e^x

f(g(x)) = e^g(x) (but g(x) = 4x)

Therefore, f(g(x)) = e^4x

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

F(x) = f(g(x)) = e^4x

We can now find the derivative of F(x) = e^4x, 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 to find the derivative of e to the 4x.

How to find the derivative of e^4x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(4)	g(x) = 4x ⇒ g'(x) = 4
	= (e^4x).4	f(g(x)) = e^4x ⇒ f'(g(x)) = e^4x
	= 4e^(4x)

Using the chain rule, the derivative of e^4x is 4e^4x

Finally, just a note on syntax and notation: the exponential function e^4x is sometimes written in the forms shown below (the derivative of each is as per the calculations above). Just be aware that not all of the forms below are mathematically correct.

e^4x	► Derivative of e^4x = 4e^4x
e^(4x)	► Derivative of e^(4x) = 4e^4x
e 4x	► Derivative of e 4x = 4e^4x
e 4 x	► Derivative of e 4 x = 4e^4x
e to the 4x	► Derivative of e to the 4x = 4e^4x

Top Tip

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

The derivative of e^ax = ae^ax

(Add the constant a to the front of the expression and keep the exponential part the same)

The Second Derivative of e^4x

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

From above, we found that the first derivative of e^4x = 4e^(4x). So to find the second derivative of e^4x, we just need to differentiate 4e^4x

We can use the chain rule to calculate the derivative of 4e^4x and get an answer of 16e^4x.

► The second derivative of e^4x = 16e^(4x)

The Derivative of e^7x

September 14, 2020 ~ DerivativeIt

The derivative of e^7x is 7e^7x

How to calculate the derivative of e^7x

In this case:

We know how to differentiate e^x (the answer is e^x)
We know how to differentiate 7x (the answer is 7)

Because e^7x is a function which is a combination of e^x and 7x, it means we can perform the differentiation of e to the 7x by making use of the chain rule.

Using the chain rule to find the derivative of e^7x

Although the function e^7x contains no parenthesis, we can still view it as a composite function (a function of a function).

If we add parenthesis around the exponent, we get e^(7x).

Now the function is in the form of the standard exponential function e^x, except it does not have x as an exponent, instead the exponent is another function of x (7x).

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

g(x) = 7x

From this it follows that:

e^7x = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = 7x, the function e^7x can be written as a composite function of f(x) and g(x).

f(x) = e^x

f(g(x)) = e^g(x) (but g(x) = 7x)

Therefore, f(g(x)) = e^7x

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

F(x) = f(g(x)) = e^7x

We can now find the derivative of F(x) = e^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 to find the derivative of e to the 7x.

How to find the derivative of e^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
	= (e^7x).7	f(g(x)) = e^7x ⇒ f'(g(x)) = e^7x
	= 7e^(7x)

Using the chain rule, the derivative of e^7x is 7e^7x

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

e^7x	► Derivative of e^7x = 7e^7x
e^(7x)	► Derivative of e^(7x) = 7e^7x
e 7x	► Derivative of e 7x = 7e^7x
e 7 x	► Derivative of e 7 x = 7e^7x
e to the 7x	► Derivative of e to the 7x = 7e^7x

The Second Derivative of e^7x

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

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

We can use the chain rule to calculate the derivative of 7e^7x and get an answer of 49e^7x.

► The second derivative of e^7x = 49e^(7x)

The Derivative of e^2x

September 11, 2020 ~ DerivativeIt

The derivative of e^2x is 2e^2x

How to calculate the derivative of e^2x

In this case:

We know how to differentiate e^x (the answer is e^x)
We know how to differentiate 2x (the answer is 2)

Because e^2x is a function which is a combination of e^x and 2x, it means we can perform the differentiation of e to the 2x by making use of the chain rule.

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

Although the function e^2x contains no parenthesis, we can still view it as a composite function (a function of a function).

If we add parenthesis around the exponent, we get e^(2x).

Now the function is in the form of the standard exponential function e^x, except it does not have x as an exponent, instead the exponent is another function of x (2x).

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

g(x) = 2x

From this it follows that:

e^2x = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = 2x, the function e^2x can be written as a composite function of f(x) and g(x).

f(x) = e^x

f(g(x)) = e^g(x) (but g(x) = 2x)

Therefore, f(g(x)) = e^2x

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

F(x) = f(g(x)) = e^2x

We can now find the derivative of F(x) = e^2x, 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 to find the derivative of e to the 2x.

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

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(2)	g(x) = 2x ⇒ g'(x) = 2
	= (e^2x).2	f(g(x)) = e^2x ⇒ f'(g(x)) = e^2x
	= 2e^(2x)

Using the chain rule, the derivative of e^2x is 2e^2x

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

e^2x	► Derivative of e^2x = 2e^2x
e^(2x)	► Derivative of e^(2x) = 2e^2x
e 2x	► Derivative of e 2x = 2e^2x
e 2 x	► Derivative of e 2 x = 2e^2x
e to the 2x	► Derivative of e to the 2x = 2e^2x

The Second Derivative of e^2x

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

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

We can use the chain rule to calculate the derivative of 2e^2x and get an answer of 4e^2x.

► The second derivative of e^2x = 4e^(2x)

The Derivative of ln(2x)

September 9, 2020 ~ DerivativeIt ~ Leave a comment

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

How to calculate the derivative of ln(2x)

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

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

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

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

In this case:

We know how to differentiate 2x (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).

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

ln(2x) 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).

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

g(x) = 2x

From this it follows that:

ln(2x) = ln(g(x))

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

f(x) = ln(x)

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

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

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

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

We can find the derivative of ln(2x) (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. 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) with respect to 2x is (1/2x).
We will use this fact as part of the chain rule to find the derivative of ln(2x) with respect to x.

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

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

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 =1/x
ln 2x	► Derivative of ln 2x = 1/x
ln 2 x	► Derivative of ln 2 x = 1/x

Finding the derivative of ln(2x) 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 2x is the product of 2 and x, we can use the product properties of logs to rewrite ln(2x):

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

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

f(x)	= ln(2) + ln(x)
f'(x)	= 0 + ln(x)	ln2 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(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) = 1/x. So to find the second derivative of ln(2x), we just need to differentiate 1/x

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

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

The Derivative of sin^2x?

September 9, 2020 ~ DerivativeIt

The derivative of sin^2x is 2sin(x)cos(x)

How to calculate the derivative of sin^2x

Note that in this post we will be looking at differentiating sin²(x) which is not the same as differentiating sin(2x). Here is our post dealing with how to differentiate sin(2x).

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

The first method is by using the product rule for derivatives (since sin²(x) can be written as sin(x).sin(x)).

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

Finding the derivative of sin^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)

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)

First, let F(x) = sin²(x)

Then remember that sin²(x) is equal to sin(x).sin(x)

So F(x) = sin(x)sin(x)

By setting f(x) and g(x) as sin(x) means that F(x) = f(x).g(x) and we can apply the product rule to find F'(x)

F'(x)	= f'(x)g(x) + f(x)g'(x)	Product Rule Definition
	= f'(x)sin(x) + sin(x)g'(x)	f(x) = g(x) = sin(x)
	= cos(x)sin(x) + sin(x)cos(x)	f'(x) = g(‘x) = cos(x)
	= 2sin(x)cos(x)

Using the product rule, the derivative of sin^2x is 2sin(x)cos(x)

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

In this case:

We know how to differentiate sin(x) (the answer is cos(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 sin^2x.

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

Although the expression sin²x contains no parenthesis, we can still view it as a composite function (a function of a function).

We can write sin²x as (sin(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 (sin(x)) as the base.

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

g(x) = sin(x)

From this it follows that:

sin(x)² = g(x)²

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

f(x) = x²

f(g(x)) = g(x)² (but g(x) = sin(x))

f(g(x)) = (sin(x))²

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

F(x) = f(g(x)) = (sin(x))²

We can find the derivative of sin^2x (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.

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

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(cos(x))	g(x) = sin(x) ⇒ g'(x) = cos(x)
	= (2sin(x)).(cos(x))	f(g(x)) = (sin(x))² ⇒ f'(g(x)) = 2sin(x)
	= 2sin(x)cos(x)

Using the chain rule, the derivative of sin^2x is 2sin(x)cos(x)

(Note – using the trigonometric identity 2cos(x)sin(x) = sin(2x), the derivative of sin^2x can also be written as sin(2x))

Finally, just a note on syntax and notation: sin^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.

sin²x	► Derivative of sin²x = 2sin(x)cos(x)
sin^2(x)	► Derivative of sin^2(x) = 2sin(x)cos(x)
sin 2 x	► Derivative of sin 2 x = 2sin(x)cos(x)
(sinx)^2	► Derivative of (sinx)^2 = 2sin(x)cos(x)
sin squared x	► Derivative of sin squared x = 2sin(x)cos(x)
sinx2	► Derivative of sinx2 = 2sin(x)cos(x)
sin^2	► Derivative of sin^2 = 2sin(x)cos(x)

The Second Derivative Of sin^2x

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

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

We can use the product rule to find the derivative of 2sin(x)cos(x).

We can set f(x) = 2sin(x) and g(x) = cos(x) and apply the product rule to find the derivative of f(x).g(x) = 2sin(x)cos(x)

The product rules says that the derivative of f(x).g(x) is equal to f’(x)g(x) + f(x)g’(x).

2cos(x)cos(x) + 2sin(x)(-sin(x))
= 2cos²(x) – 2sin²(x)
= 2(cos²(x) – sin²(x))

Using the trigonometric double angle identity cos(2x) = cos²(x) – sin²(x), we can rewrite this as

= 2cos(2x)

► The second derivative of sin^2x is 2cos(2x)

Interestingly, the second derivative of sin²x is equal to the first derivative of sin(2x).

The Derivative of e^3x

August 5, 2020 ~ DerivativeIt

The derivative of e^3x is 3e^3x

How to calculate the derivative of e^3x

In this case:

We know how to differentiate e^x (the answer is e^x)
We know how to differentiate 3x (the answer is 3)

Because e^3x is a function which is a combination of e^x and 3x, it means we can perform the differentiation of e to the 3x by making use of the chain rule.

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

Although the function e^3x contains no parenthesis, we can still view it as a composite function (a function of a function).

If we add parenthesis around the exponent, we get e^(3x).

Now the function is in the form of the standard exponential function e^x, except it does not have x as an exponent, instead the exponent is another function of x (3x).

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

g(x) = 3x

From this it follows that:

e^3x = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = 3x, the function e^3x can be written as a composite function of f(x) and g(x).

f(x) = e^x

f(g(x)) = e^g(x) (but g(x) = 3x)

Therefore, f(g(x)) = e^3x

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

F(x) = f(g(x)) = e^3x

We can now find the derivative of F(x) = e^3x, 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 to find the derivative of e to the 3x.

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

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(3)	g(x) = 3x ⇒ g'(x) = 3
	= (e^3x).3	f(g(x)) = e^3x ⇒ f'(g(x)) = e^3x
	= 3e^(3x)

Using the chain rule, the derivative of e^3x is 3e^3x

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

e^3x	► Derivative of e^3x = 3e^3x
e^(3x)	► Derivative of e^(3x) = 3e^3x
e 3x	► Derivative of e 3x = 3e^3x
e 3 x	► Derivative of e 3 x = 3e^3x
e to the 3x	► Derivative of e to the 3x = 3e^3x

The Second Derivative of e^3x

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

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

We can use the chain rule to calculate the derivative of 3e^3x and get an answer of 9e^3x.

► The second derivative of e^3x = 9e^(3x)