Question

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

Answer 1

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(x) = y = ln(x^2 + x)/ln(10)`

We can consider `1/ln10` to just be a constant, so take the derivative of the numerator and apply the chain rule:

`dy/dx = 1/ln(10) * 1/(x^2 + x) * (2x + 1)`

Simplify a bit:

`dy/dx = (2x + 1)/(ln(10)*(x^2 + x))`

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.