# How do you find f'(x) for f(x) = (ln x)^8?

Jul 29, 2015

This shall be accomplished by the chain rule.

#### Explanation:

Let, $y = f \left(x\right)$ where $f \left(x\right) = {\left(L n x\right)}^{8}$ .

We have to evaluate $\frac{\mathrm{dy}}{\mathrm{dx}}$.
Now, let $t = L n x \implies y = {t}^{8}$

Now, let us differentiate $y$ with respect to $x$,

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{dt}} \cdot \frac{\mathrm{dt}}{\mathrm{dx}}$

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = 8 {t}^{7} \cdot \frac{d}{\mathrm{dx}} \left(L n x\right)$

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = 8 {\left(L n x\right)}^{7} / x$, which is the derivative we were looking for.