# Derivative of sin x

How to solve the derivative of sin x problems: formula, proof, examples, and their solutions.

## Formula

The derivative of sin x is cos x.

## Proof

Write the gray term
at the front part.

And combine the sin x terms.

Split the fraction into two parts.

Write the denominators h
under (sin h) and (cos h - 1).

Recall that as h → 0,

(sin h)/h → 1.

But (cos h - 1)/h is 0/0 form.

So multiply the conjugate of (cos h - 1), (cos h + 1)
to both of the numerator and the denominator.

Limits of trigonometric functions

sin2 h + cos2 h = 1

So cos2 h - 1 = -sin2 h.

Pythagorean identities

Change -sin2 h into sin h.

And write -sin h
on the numerator of 1/(cos h + 1).

As h → 0,

(sin h)/h → 1 (gray)
and -sin h → 0. (red)

So the limit is
cos x⋅1 + sin x⋅1⋅(-0/(1 + 1)),
which is cos x.

So [sin x]' = cos x.

## Example 1

y = (x)(sin x)

So y' is equal to,
the derivative of x, 1
times sin x
plus x,
times, the derivative of sin x, cos x.

Product rule in differentiation

Power rule in differentiation (Part 1)

## Example 2

y = sin (x2)

So y' is equal to,
the derivative of the outer part, cos (x2)
times, the derivative of the inner part, 2x.

Chain rule in differentiation

Power rule in differentiation (Part 1)

## Example 3

The given function is slightly different
from the previous example's function.

y = (sin x)2

So y' is equal to,
the derivative of the outer part, 2(sin x)
times, the derivative of the inner part, cos x.

Chain rule in differentiation

Power rule in differentiation (Part 1)