Derivative of sin x

Derivative of sin x

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

Formula

[sin x]' = cos x

The derivative of sin x is cos x.

Proof

Derivative of sin x: Proof of the Formula

Definition of a derivative function

sin (A + B)

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

Find the derivative of the given function. y = x sin x

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

Find the derivative of the given function. y = sin x^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

Find the derivative of the given function. y = sin^2 x

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)