Question

How do you find the derivative of `sinh^(79x)`?

Answer 1

If it is operator `(sinh)^79`, operating oh the operand x and `f_79=((sinh)^79)x`, the derivative `f'_79` is the product `(cosh f_78) (cosh f_77)(cosh f_76)...(cosh f_1)(cosh x)`

Explanation:

Let `f_79=(sinh^79)(x)`.

Then `f_79=sinh (f_78)`.

It follows that

`f'_79=cosh (f_78) f'_78`, using function pf function rule.

This is a reduction formula.

Successive reduction leads to

`f'_79=(cosh f_78) (cosh f_77)(cosh f_76)...(cosh f_1)(cosh x)`

`f_1=sinh x. f'_1=cosh x`. .