Find f'(x) if f(x)=5^tan5x.

f(x) = 5^(tan5x)

We will use
the chain rule to find the derivative.

Let g(x) =
tan5x.

==> g'(x) = 5*sec^2
(5x)

==> f(x) =
5^g(x).

Let us differentiate using the chain
rule.

==> f'(x) = [5^(g(x))' *
g'(x).

             = (5^(tan5x))' *
(tan5x)'

            = (5^tan5x * ln 5 )* (5sec^2
5x)

==> f'(x) = 5*ln 5 (5^tan5x) * sec^2
5x

But we know that sec^2 5x = 1/cos^2
5x

==> f'(x) = (5^tan5x + 1)*ln 5 /
cos^2 5x.