Derivative of a Complicated Power Equation

Calculus Level pending

Let $f\left(x\right)=a^{\left(a^{x}\right)}$ and $g\left(x\right)=log_{a} {x}$ where $a$ is a constant.

What is the derivative of $g\left(g\left(f\left(x\right)\right)\right)$ ?

1 solution

Tristan Shin
Jan 4, 2014

First, find $g\left(g\left(f\left(x\right)\right)\right)$ .

$g\left(f\left(x\right)\right)=log_{a} {a^{\left(a^{x}\right)}=a^{x}}$

$g\left(g\left(f\left(x\right)\right)\right)=log_{a} {a^{x}}=x$

Therefore, $\frac{d}{dx} {g\left(g\left(f\left(x\right)\right)\right)}=1$ .