Integral of cot^2(x) (trigonometric identity)
Integral of cot^2(x) (trigonometric identity)

The derivative of cot(2x) is -2csc2(2x)

How to calculate the derivative of cot(2x)

Note that in this post we will be looking at differentiating cot(2x) which is not the same as differentiating cot2(x). Here is our post dealing with how to differentiate cot^2(x).

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 cosec2(x))
  • We know how to differentiate 2x (the answer is 2)

This means the chain rule will allow us to differentiate the expression cot(2x).

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

To perform the differentiation cot(2x), 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 is actually in terms of (in this case the derivative of 2x).

Let’s call the function in the argument of cot, g(x), which means the function is in the form of cot(x), except it does not have x as the angle, instead it has another function of x (2x) as the angle

If:

g(x) = 2x

It follows that:

cot(2x) = cot(g(x))

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

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

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

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

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. But before we do that, just a quick recap on the derivative of the cot function.

The derivative of cot(x) with respect to x is -csc2(x)
The derivative of cot(z) with respect to z is -csc2(z)

In a similar way, the derivative of cot(2x) with respect to 2x is -csc2(2x).

We will use this fact as part of the chain rule to find the derivative of cot(2x) with respect to x.

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))(2) g(x) = 2x ⇒ g'(x) = 2
= (-csc2(2x)).(2) f(g(x)) = cot(2x) ⇒ f'(g(x)) = -cosec2(2x)
= -2csc2(2x)

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

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

cot2x ► Derivative of cot2x = -2csc2(2x)
cot 2 x ► Derivative of cot 2 x = -2csc2(2x)
cot 2x ► Derivative of cot 2x = -2csc2(2x)
cot (2x) ► Derivative of cot (2x) = -2csc2(2x)

The Second Derivative Of cot(2x)

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

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

We can use the chain rule to find the derivative of -2csc(2x) (bearing in mind that the derivative of csc^2(x) is -2csc2(x)cot(x)) and it gives us a result of 8csc2(2x)cot(2x)

► The second derivative of cot(2x) is 8csc2(2x)cot(2x)

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