Derivative of cos x

Derivative of cos x

How to solve the derivative of cos x problems: formula, proof, example, and its solution.

Formula

[cos x]' = -sin x

The derivative of cos x is -sin x.

Proof

Derivative of cos x: Proof of the Formula

cos x = sin (π/2 - x)

Trigonometric functions of (90º - θ)

Think sin (π/2 - x) as a composite function.

Then [sin (π/2 - x)]' is equal to,
the derivative of the outer part, cos (π/2 - x)
times, the derivative of the inner part, -1.

Chain rule in differentiation

Derivatives of polynomials

cos (π/2 - x) = sin x

Trigonometric functions of (90º - θ)

So [cos x]' = -sin x.

Example

Find the derivative of the given function. y = cos (x^3 - 4)

y = cos (x3 - 4)

So y' is equal to,
the derivative of the outer part, -sin (x3 - 4)
times, the derivative of the inner part, 3x2.

Chain rule in differentiation

Derivatives of polynomials