The derivative of arc(cotx) is equal to -1/(1+x^{2}). 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+x ^{2}). **

**Substitution Method:**

Let y = arc cotx.

So we need to find dy/dx.

Note that we have y = cot^{-1}x

⇒ coty = x **…(I)**

Differentiating both sides with respect to x, we get that

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

⇒ – cosec^{2}y $\dfrac{dy}{dx}$ =1

⇒ $\dfrac{dy}{dx}$ = -1/cosec^{2}y

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

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

Thus, the derivative of arc cotx, i.e., the derivative of cot inverse x is equal to – 1/(1+x^{2}).

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

We know that

tan^{-1}x + cot^{-1}x = π/2

Take the differentiation on both sides, so we have

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

⇒ 1/(1+x^{2}) + $\dfrac{d}{dx}$(cot^{-1}x) =0

⇒ $\dfrac{d}{dx}$(cot^{-1}x) = -1/(1+x^{2}).

Thus, we have shown that the derivative of arccot x is -1/(1+x^{2}).

## FAQs

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

Answer: If y=arc cotx, then dy/dx = -1/(1+x^{2}).