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

Asked on by abigaile

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

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.

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To determine the derivative of the given composed function, we'll differentiate both sides:

dy= [5^ (tan 5x)]'dx

We'll apply chain rule:

dy = 5^ (tan 5x)*ln 5*(tan 5x)'dx

But (tan 5x)' = (5x)'/(cos 5x)^2

(tan 5x)' = 5/(cos 5x)^2

dy = (5*5^ (tan 5x)*ln 5)/(cos 5x)^2 dx

We'll add the exponents of 5:

dy/dx = (5^ (tan 5x + 1)*ln 5)/(cos 5x)^2 dx

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