Derivative of sinh x

Derivative of sinh x

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

Formula

[sinh x]' = cosh x

The derivative of sinh x is cosh x.

Proof

Derivative of sinh x: Proof of the Formula

sinh x = (ex - e-x) / 2

Hyperbolic functions

[ex]' = ex

Derivative of ex

[e-x]' = (e-x)⋅(-1)

Chain rule in differentiation

(ex + e-x) / 2 = cosh x

So [sinh x]' = cosh x.

Example

Find the derivative of the given function. y = sinh x^3

y = sinh (x3)

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

Chain rule in differentiation

Power rule in differentiation (Part 1)