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

Formula

The derivative of sin x, [sin x]',
is cos x.

Example 1

Example

Solution

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.

Example 2

Example

Solution

y = sin x2 is a composite function of
y = sin (whole) and (whole) = x2.

So the derivative of the composite function is,
the derivative of the outer function sin x2, cos x2
times,
the derivative of the inner function x2, 2x1.

(cos x2)⋅(2x1) = 2x cos x2

So 2x cos x2 is the derivative of sin x2.

Example 3

Example

Solution

y = sin2 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 sin2 x.