# How do you differentiate f(x)=ln(3x^2) using the chain rule?

Dec 28, 2015

Step by step explanation and working is given below.

#### Explanation:

$f \left(x\right) = \ln \left(3 {x}^{2}\right)$

For chain rule, first break the problem into smaller links and find their derivatives. The final answer would be the product of all the derivatives in the link.

$y = \ln \left(u\right)$
$u = 3 {x}^{2}$

The differentiation using chain rule would be

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

$y = \ln \left(u\right)$
Differentiate with respect to $u$

$\frac{\mathrm{dy}}{\mathrm{du}} = \frac{1}{u}$

$u = 3 {x}^{2}$
Differentiate with respect to $x$
$\frac{\mathrm{du}}{\mathrm{dx}} = 3 \frac{d \left({x}^{2}\right)}{\mathrm{dx}}$
$\frac{\mathrm{du}}{\mathrm{dx}} = 3 \cdot 2 x$
$\frac{\mathrm{du}}{\mathrm{dx}} = 6 x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{u} \cdot 6 x$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6 x}{u}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6 x}{3 {x}^{2}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2}{x}$ Final answer

Note: The above process might look big, this was given in such a form to help you understand step by step working. With practice, you can do it quickly and with less steps.