# Derivative of sin x

How to find the derivative of the given function by using the derivative of sin x: formula, 3 examples, and their solutions.

## Formula

The derivative of sin x, [sin x]',

is cos x.

## Exampley = x sin x, y' = ?

x sin x

is the product of x and sin x.

So the derivative of the product is,

the derivative of x, 1

times sin x

plus

x

times, the derivative of sin x, cos x.

1⋅sin x = sin x

So

sin x + x cos x

is the derivative of x sin x.

## Exampley = sin x^{2}, y' = ?

y = sin x^{2}

is a composite function of

y = sin (whole) and (whole) = x^{2}.

So the derivative of the composite function is,

the derivative of the outer function sin x^{2}, cos x^{2}

times,

the derivative of the inner function x^{2}, 2x^{1}.

(cos x^{2})⋅(2x^{1}) = 2x cos x^{2}

So

2x cos x^{2}

is the derivative of sin x^{2}.

## Exampley = sin^{2} x, y' = ?

y = sin^{2} x

means

y = (sin x)^{2}.

This is a composite function of

y = (whole)^{2} and (whole) = sin x.

So the derivative of the composite function is,

the derivative of the outer function (sin x)^{2}, 2(sin x)^{1}

times,

the derivative of the inner function sin x, cos x.

2(sin x)^{1}⋅cos x = 2 sin x cos x

Then, by the sin 2A formula,

2 sin x cos x = sin 2x.

So

sin 2x

is the derivative of sin^{2} x.