# Differentiation of sinx

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

Let’s begin –

## Differentiation of sinx

The differentiation of sinx with respect to x is cosx.

i.e. $$d\over dx$$ (sinx) = cosx

## Proof Using First Principle :

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

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

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

By using trigonometry formula,

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

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

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

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

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

$$\implies$$ $$d\over dx$$(f(x)) = (cos x) $$\times$$ 1 = cos x

Hence, $$d\over dx$$ (sin x) = cos x

Example : What is the differentiation of sin 2x – 2 sin x with respect to x?

Solution : Let y = sin 2x – 2 sin x

$$d\over dx$$(y) = $$d\over dx$$(sin 2x – 2 sin x)

$$\implies$$ $$d\over dx$$(y) = $$d\over dx$$(sin 2x) – $$d\over dx$$(2 sinx)

By using chain rule and sinx differentiation we get,

$$\implies$$ $$d\over dx$$(y) = 2 cos 2x + 2 $$d\over dx$$(sinx)

$$\implies$$ $$d\over dx$$(y) = 2 cos 2x + 2 cos x

Hence, $$d\over dx$$(sin 2x – 2 sin x) = 2 cos 2x + 2 cos x

Example : What is the differentiation of $$x^2$$ +  sin x with respect to x?

Solution : Let y = $$x^2$$ +  sin x

$$d\over dx$$(y) = $$d\over dx$$($$x^2$$ +  sin x)

$$\implies$$ $$d\over dx$$(y) = $$d\over dx$$$$x^2$$ – $$d\over dx$$(sinx)

By using differentiation formulas we get,

$$\implies$$ $$d\over dx$$(y) = 2x + cos x

Hence, $$d\over dx$$($$x^2$$ +  sin x) = 2x + cos x

### Related Questions

What is the Differentiation of sin inverse x ?

What is the differentiation of $$sin x^2$$ ?

What is the Integration of sin x ?