Question

How do you differentiate `f(x)=tan(5x)`?

Answer 1

`f'(x)=5sec^2(5x)`

Explanation:

Since the function is composed of a function in terms of another function, chain rule must be applied. The two functions are `tan(5x)`, which is in terms of `5x`, and `5x` which is in terms of `x`.

`f'(x)=(dy)/(du)*(du)/(dx)`

Let `u=5x`

`(dy)/(du)=d/(du)tan(u)=sec^2(u)`

`(du)/(dx)=d/(dx)5x=5`

`:.f'(x)=5*sec^2(u)=5*sec^2(5x)`

`=5sec^2(5x)`