Find the Derivative of y = cot(3x^2 + 5)
Find the Derivative of y = cot(3x^2 + 5)

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)

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