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).


Q1: If y=arc cotx, then find dy/dx?

Answer: If y=arc cotx, then dy/dx = -1/(1+x2).

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