Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 68

Find a.) $F'(x)$ and b.) $G'(x)$ where $F(x) = f(x^{\alpha}$ and $G(x) = [f(x)^{\alpha}]$. Suppose that $f$ is differentiable everywhere and $\alpha$ is a real number.


$
\begin{equation}
\begin{aligned}

\text{ a.) } F'(x) = \frac{d}{dx} [f(x)] =& f'(x^{\alpha}) \cdot \alpha (x^{\alpha - 1})
\\
\\
F'(x) = \frac{d}{dx} [f(x)] =& \alpha x^{\alpha - 1} f'(x^{\alpha})
\\
\\
\\
\text{ b.) } G'(x) = \frac{d}{dx} [G(x) ] =& \alpha [f(x)]^{\alpha - 1} \cdot f'(x) \cdot 1
\\
G'(x) = \frac{d}{dx} [G(x) ] =& \alpha [f(x)]^{\alpha - 1} \cdot f'(x) \cdot 1
\\
\\
G'(x) = \frac{d}{dx} [G(x) ] =& \alpha [f(x)]^{\alpha - 1} f'(x)

\end{aligned}
\end{equation}
$