The Derivative of cot^2x

October 5, 2020 ~ DerivativeIt

The derivative of cot^2x is -2.csc^2(x).cot(x)

How to calculate the derivative of cot^2x

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

The first method is by using the product rule for derivatives (since cot²(x) can be written as cot(x).cot(x)).

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

Finding the derivative of cot^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) = cot²(x)

Then remember that cot²(x) is equal to cot(x).cot(x)

So F(x) = cot(x)cot(x)

By setting f(x) and g(x) as cot(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)cot(x) + cot(x)g'(x)	f(x) = g(x) = cot(x)
	= -csc²(x)cot(x) + cot(x)(-csc²(x))	f'(x) = g(‘x) = -csc²(x)
	= -2csc²(x)cot(x)

Using the product rule, the derivative of cot^2x is -2csc²(x)cot(x)

Finding the derivative of cot^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 cot(x) (the answer is -csc²(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 cot^2x.

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

Although the expression cot²x contains no parenthesis, we can still view it as a composite function (a function of a function).

We can write cot²x as (cot(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 (cot(x)) as the base.

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

g(x) = cot(x)

From this it follows that:

(cot(x))² = g(x)²

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

f(x) = x²

f(g(x)) = g(x)² (but g(x) = cot(x))

f(g(x)) = (cot(x))²

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

F(x) = f(g(x)) = (cot(x))²

We can find the derivative of cot^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 cot^2x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(-csc²(x))	g(x) = cot(x) ⇒ g'(x) = -csc²(x)
	= (2.cot(x)).(-csc²(x))	f(g(x)) = (cot(x))² ⇒ f'(g(x)) = 2.cot(x)
	= -2csc²(x)cot(x)

Using the chain rule, the derivative of cot^2x is -2.csc²(x)cot(x)

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

cot²x	► Derivative of cot²x = -2.csc²(x).cot(x)
cot^2(x)	► Derivative of cot^2(x) = -2.csc²(x).cot(x)
cot 2 x	► Derivative of cot 2 x = -2.csc²(x).cot(x)
(cotx)^2	► Derivative of (cotx)^2 = -2.csc²(x).cot(x)
cot squared x	► Derivative of cot squared x = -2.csc²(x).cot(x)
cotx2	► Derivative of cotx2 = -2.csc²(x).cot(x)
cot^2	► Derivative of cot^2 = -2.csc²(x).cot(x)

The Second Derivative Of cot^2x

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

From above, we found that the first derivative of cot^2x = -2csc²(x)cot(x). So to find the second derivative of cot^2x, we need to differentiate -2csc²(x)cot(x).

We can use the product and chain rules, and then simplify to find the derivative of -2csc²(x)cot(x) is 4csc²(x)cot²(x) + 2csc⁴(x)

► The second derivative of cot^2x is 4csc²(x)cot²(x) + 2csc⁴(x)

Interesting property of the derivative of cot^2x

It is interesting to note that the derivative of cot²x is equal to the derivative of csc²x.

The derivative of:
> cot²x = -2.csc²(x).cot(x)
> csc²x = -2.csc²(x).cot(x)

The Derivative of csc^2x

October 5, 2020 ~ DerivativeIt

The derivative of csc^2x is -2.csc^2(x).cot(x)

How to calculate the derivative of csc^2x

Note that in this post we will be looking at differentiating csc²(x) which is not the same as differentiating csc(2x). Here is our post dealing with how to differentiate csc(2x).

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

The first method is by using the product rule for derivatives (since csc²(x) can be written as csc(x).csc(x)).

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

Finding the derivative of csc^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) = csc²(x)

Then remember that csc²(x) is equal to csc(x).csc(x)

So F(x) = csc(x)csc(x)

By setting f(x) and g(x) as csc(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)csc(x) + csc(x)g'(x)	f(x) = g(x) = csc(x)
	= -csc(x)cot(x)csc(x) + csc(x)(-csc(x)cot(x))	f'(x) = g(‘x) = -csc(x)cot(x)
	= -2csc²(x)cot(x)

Using the product rule, the derivative of csc^2x is -2csc^2(x)cot(x)

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

In this case:

We know how to differentiate csc(x) (the answer is -csc(x)cot(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 csc^2x.

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

Although the expression csc²x contains no parenthesis, we can still view it as a composite function (a function of a function).

We can write csc²x as (csc(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 (csc(x)) as the base.

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

g(x) = csc(x)

From this it follows that:

csc(x)² = g(x)²

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

f(x) = x²

f(g(x)) = g(x)² (but g(x) = csc(x))

f(g(x)) = (csc(x))²

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

F(x) = f(g(x)) = (csc(x))²

We can find the derivative of csc^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 csc^2x using the Chain Rule:

F'(x)	= f'(g(x)).g'(x)	Chain Rule Definition
	= f'(g(x))(-csc(x)cot(x))	g(x) = csc(x) ⇒ g'(x) = -csc(x)cot(x)
	= (2csc(x)).(csc(x)tan(x))	f(g(x)) = (csc(x))² ⇒ f'(g(x)) = 2csc(x)
	= -2csc²(x)cot(x)

Using the chain rule, the derivative of csc^2x is -2.csc^2(x).cot(x)

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

csc²x	► Derivative of csc²x = -2csc²(x)cot(x)
csc^2(x)	► Derivative of csc^2(x) = -2csc²(x)cot(x)
csc 2 x	► Derivative of csc 2 x = -2csc²(x)cot(x)
(cscx)^2	► Derivative of (cscx)^2 = -2csc²(x)cot(x)
csc squared x	► Derivative of csc squared x = -2csc²(x)cot(x)
cscx2	► Derivative of cscx2 = -2csc²(x)cot(x)
csc^2	► Derivative of csc^2 = -2csc²(x)cot(x)

The Second Derivative Of csc^2x

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

From above, we found that the first derivative of csc^2x = -2csc²(x)cot(x). So to find the second derivative of csc^2x, we need to differentiate -2csc²(x)cot(x).

We can use the product and chain rules, and then simplify to find the derivative of -2csc²(x)cot(x) is 4csc²(x)cot²(x) + 2csc⁴(x)

► The second derivative of csc^2x is 4csc²(x)cot²(x) + 2csc⁴(x)

Interesting property of the derivative of csc^2x

It is interesting to note that the derivative of csc²x is equal to the derivative of cot²x.

The derivative of:
> csc²x = -2.csc²(x).cot(x)
> cot²x = -2.csc²(x).cot(x)

The Derivative of e^8x

October 5, 2020 ~ DerivativeIt

The derivative of e^8x is 8e^8x

How to calculate the derivative of e^8x

In this case:

We know how to differentiate e^x (the answer is e^x)
We know how to differentiate 8x (the answer is 8)

Because e^8x is a function which is a combination of e^x and 8x, it means we can perform the differentiation of e to the 8x by making use of the chain rule.

Using the chain rule to find the derivative of e^8x

Although the function e^8x 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^(8x).

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 (8x).

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

g(x) = 8x

From this it follows that:

e^8x = e^g(x)

Let’s set f(x) = e^x.

Then, because g(x) = 8x, the function e^8x 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) = 8x)

Therefore, f(g(x)) = e^8x

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

F(x) = f(g(x)) = e^8x

We can now find the derivative of F(x) = e^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 to find the derivative of e to the 8x.

How to find the derivative of e^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
	= (e^8x).8	f(g(x)) = e^8x ⇒ f'(g(x)) = e^8x
	= 8e^(8x)

Using the chain rule, the derivative of e^8x is 8e^8x

Finally, just a note on syntax and notation: the exponential function e^8x 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^8x	► Derivative of e^8x = 8e^8x
e^(8x)	► Derivative of e^(8x) = 8e^8x
e 8x	► Derivative of e 8x = 8e^8x
e 8 x	► Derivative of e 8 x = 8e^8x
e to the 8x	► Derivative of e to the 8x = 8e^8x

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^8x

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

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

We can use the chain rule to calculate the derivative of 8e^8x and get an answer of 64e^8x.

► The second derivative of e^8x = 64e^(8x)