# Differentiation of cotx

Here you will learn what is the differentiation of cotx and its proof by using first principle.

Let’s begin –

## Differentiation of cotx

The differentiation of cotx with respect to x is $$-cosec^2x$$.

i.e. $$d\over dx$$ (cotx) = $$-cosec^2x$$

## Proof Using First Principle :

Let f(x) = cot x. Then, f(x + h) = cot(x + h)

$$\therefore$$   $$d\over dx$$(f(x)) = $$lim_{h\to 0}$$ $$f(x + h) – f(x)\over h$$

$$d\over dx$$(f(x)) = $$lim_{h\to 0}$$ $$cot(x + h) – cot x\over h$$

$$\implies$$  $$d\over dx$$(f(x)) = $$lim_{h\to 0}$$ $${cos(x + h)\over sin(x + h)} – {cos x\over sin x}\over h$$

$$\implies$$  $$d\over dx$$(f(x)) = $$lim_{h\to 0}$$ $$sin x cos(x + h)- cos x sin(x + h)\over h sin x sin(x +h)$$

By using trigonometry formula,

[sin A cos B – cos A sin B = sin (A – B)]

$$\implies$$ $$d\over dx$$(f(x)) = $$lim_{h\to 0}$$ $$sin h\over h$$.$$1\over sin x sin (x + h)$$

$$\implies$$ $$d\over dx$$(f(x)) = -$$lim_{h\to 0}$$ $$sin h\over h$$ $$lim_{h\to 0}$$$$1\over sin x sin (x + h)$$

because, [$$lim_{h\to 0}$$$$sin(h/2)\over (h/2)$$ = 1]

$$\implies$$ $$d\over dx$$(f(x)) = -1.$$1\over sin x sin x$$ = $$-cosec^2x$$

Hence, $$d\over dx$$ (cot x) = $$-cosec^2x$$

Example : What is the differentiation of cot x + 1 with respect to x?

Solution : Let y = cot x + 1

$$d\over dx$$(y) = $$d\over dx$$(cot x + 1)

$$\implies$$ $$d\over dx$$(y) = $$d\over dx$$(cot x) + $$d\over dx$$(1)

By using cotx differentiation we get,

$$\implies$$ $$d\over dx$$(y) = $$-cosec^2x$$ + 0

Hence, $$d\over dx$$(cot x + 1) = $$-cosec^2x$$

Example : What is the differentiation of $$cot\sqrt{x}$$ with respect to x?

Solution : Let y = $$cot\sqrt{x}$$

$$d\over dx$$(y) = $$d\over dx$$($$cot\sqrt{x}$$)

By using chain rule we get,

$$\implies$$ $$d\over dx$$(y) = $$1\over 2\sqrt{x}$$($$-cosec^2\sqrt{x}$$)

Hence, $$d\over dx$$($$cot\sqrt{x}$$) = -$$1\over 2\sqrt{x}$$$$cosec^2\sqrt{x}$$

### Related Questions

What is the Differentiation of cot inverse x ?

What is the Differentiation of cosx ?

What is the Integration of Cot x ?