How to find Derivative of arc(cotx)

The derivative of arc(cotx) is equal to -1/(1+x²). In this post, we will learn how to find the derivative of arc cotx.

Question: What is the derivative of arc cotx?

Answer: The derivative of arccotx is -1/(1+x²).

Substitution Method:

Let y = arc cotx.

So we need to find dy/dx.

Note that we have y = cot^-1x

⇒ coty = x …(I)

Differentiating both sides with respect to x, we get that

$\dfrac{d}{dx}$(coty) = $\dfrac{dx}{dx}$

⇒ – cosec²y $\dfrac{dy}{dx}$ =1

⇒ $\dfrac{dy}{dx}$ = -1/cosec²y

⇒ $\dfrac{dy}{dx}$ = – $\dfrac{1}{1+\cot^2 y}$ as we know that cosec²y = 1+cot²y

⇒ $\dfrac{dy}{dx}$ = – $\dfrac{1}{1+x^2}$ as coty = x by (I)

Thus, the derivative of arc cotx, i.e., the derivative of cot inverse x is equal to – 1/(1+x²).

Method2: (Using the derivative of tan inverse x):

We know that

tan^-1x + cot^-1x = π/2

Take the differentiation on both sides, so we have

$\dfrac{d}{dx}$(tan^-1x) + $\dfrac{d}{dx}$(cot^-1x) = 0 since the derivative of a constant is zero and π/2 is a constant.

⇒ 1/(1+x²) + $\dfrac{d}{dx}$(cot^-1x) =0

⇒ $\dfrac{d}{dx}$(cot^-1x) = -1/(1+x²).

Thus, we have shown that the derivative of arccot x is -1/(1+x²).

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