n -th derivative of $f(t)^{\alpha}$

Question

Im looking for a closed expresion for the $n$-th derivative of $f(t)^{\alpha}$ with $\alpha\in\mathbb{R}$, that is:

$$\frac{d^n}{d\,t^n}f(t)^{\alpha}$$

Answer 1

$\dfrac{d}{dt}(f(t)^{\alpha})=\alpha f(t)^{\alpha -1}f'(t)$

$\dfrac{d^2}{dt^2}(f(t)^{\alpha})=\alpha[f(t)^{\alpha -1}f'(t)]'=\alpha[(\alpha-1)f(t)^{\alpha -2}(f'(t))^2+f(t)^{\alpha -1}f''(t)]$

$\dfrac{d^3}{dt^3}(f(t)^{\alpha})=\alpha[(\alpha-1)(\alpha-2)f(t)^{\alpha -3}(f'(t))^3+2(\alpha - 1)f(t)^{\alpha -2}(f'(t))^2+(\alpha - 1)f(t)^{\alpha-1}f'(t)f''(t)+f(t)^{\alpha-1}f'''(t)]$