# How do you differentiate f(x)=ln(x^2) using the chain rule?

May 29, 2016

Take the derivative of ${x}^{2}$ and divide it by ${x}^{2}$ to get $f ' \left(x\right) = \frac{2}{x}$.

#### Explanation:

The chain rule tells us that the derivative of a compound function - like $\ln \left({x}^{2}\right)$, which is made up of two functions ($\ln x$ and ${x}^{2}$), is the derivative of the whole thing times the derivative of the inside function. In math terms:
$f \left(g \left(x\right)\right) ' = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

As it applies to the natural log function, the chain rule says:
$\ln \left(u\right) ' = u ' \cdot \frac{1}{u} = \frac{u '}{u}$
Where $u$ is a function of $x$.

In our case ($f \left(x\right) = \ln \left({x}^{2}\right)$), $u = {x}^{2}$, so the derivative is:
$f ' \left(x\right) = \frac{{x}^{2} '}{{x}^{2}} = \frac{2 x}{x} ^ 2 = \frac{2}{x}$