# What is the derivative of f(x)=log(x^2+x) ?

Jul 31, 2014

I'll assume that by $\log$ you meant a logarithm with base 10. Shouldn't be an issue anyways since the logic applies to other bases as well.

First we will apply the change-of-base rule:

$f \left(x\right) = y = \ln \frac{{x}^{2} + x}{\ln} \left(10\right)$

We can consider $\frac{1}{\ln} 10$ to just be a constant, so take the derivative of the numerator and apply the chain rule:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{\ln} \left(10\right) \cdot \frac{1}{{x}^{2} + x} \cdot \left(2 x + 1\right)$

Simplify a bit:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x + 1}{\ln \left(10\right) \cdot \left({x}^{2} + x\right)}$

There's our derivative. Keep in mind, taking derivatives of logarithms without base $e$ is just a matter of using change-of-base rule to convert them to natural logarithms, which are easy to differentiate.