# Differentiation of cosecx

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

Let’s begin –

## Differentiation of cosecx

The differentiation of cosecx with respect to x is -cosecx.cotx

i.e. $$d\over dx$$ (cosecx) = -cosecx.cotx

## Proof Using First Principle :

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

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

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

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

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

By using trigonometry formula,

[sin C – sin D = $$2sin ({C – D\over 2})cos ({C + D\over 2})$$]

$$\implies$$ $$d\over dx$$(f(x)) = $$lim_{h\to 0}$$ $$2sin ({x – x – h\over 2})cos({x + x + h\over 2})\over h sin x sin (x + h)$$

$$\implies$$ $$d\over dx$$(f(x)) = $$lim_{h\to 0}$$ $$2sin ({-h\over 2})cos({x + h/2})\over h sin x sin (x + h)$$

$$\implies$$ $$d\over dx$$(f(x)) = -$$lim_{h\to 0}$$ $$cos ({x + h/2})\over sin x sin(x + h)$$.$$lim_{h\to 0}$$ $$sin(h/2)\over (h/2)$$

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

$$\implies$$ $$d\over dx$$(f(x)) = -$$cos x\over sin x sin x$$(1) = -cot x cosec x

Hence, $$d\over dx$$ (cosec x) = =cosecx.cotx

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

Solution : Let y = cosec x + x

$$d\over dx$$(y) = $$d\over dx$$(cosec x + x)

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

By using cosecx differentiation we get,

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

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

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

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

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

By using chain rule we get,

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

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

### Related Questions

What is the Differentiation of cosec inverse x ?

What is the Integration of Cosecx ?

What is the differentiation of 1/sinx ?