Derivative of cosh x

Derivative of cosh x

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

Formula

[cosh x]' = sinh x

The derivative of cosh x is sinh x.

Unlike [cos x]' = -sin x,
the sign doesn't change.

Derivative of cos x

Proof

Derivative of cosh x: Proof of the Formula

cosh 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 = sinh x

So [cosh x]' = sinh x.

Example

Find the derivative of the given function. y = 1/cosh x

y = 1 / (cosh x)

So y' is equal to,
the derivative of cosh x, sinh x
over cosh2 x.

Reciprocal rule in differentiation

sinh x/cosh x = tanh x
1/cosh x = sech x

So (given) = -tanh x sech x.