Derivative of sinh x

How to find the derivative of the given function by using the derivative of sinh x: definition, formula, 1 example, and its solution.

Hyperbolic Function

Definition

The hyperbolic functions are functions
that look like trigonometric functions
with the letter h:
sinh x, cosh x, tanh x, ... .

sinh x = (ex - e-x)/2
It's read as [hyperbolic sine] or [sin-ch].

cosh x = (ex + e-x)/2
It's read as [hyperbolic cosine] or [co-sh].

There's a reason these are called hyperbolic functions.

Recall that
a point (cos θ, sin θ)
is always on the unit circle x2 + y2 = 1.

Similar to this,
a point ((ex + e-x)/2, (ex - e-x)/2)
is always on the unit hyperbola x2 - y2 = 1.

So (cosh x, sinh x) is defined as ((ex + e-x)/2, (ex - e-x)/2).

This is why sinh x = (ex - e-x)/2 and cosh x = (ex + e-x)/2.

Formula

Formula

The derivative of sinh x, [sinh x]',
is cosh x.

Example

Example

Solution

y = sinh x5 is a composite function of
y = sinh (whole) and (whole) = x5.

So the derivative of the composite function is,
the derivative of the outer function sinh x5, cosh x5
times,
the derivative of the inner function x5, 5x4.

Arrange the expression.

So 5x4 cosh x5 is the derivative of sinh x5.