Question

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

Answer 1

Take the derivative of `x^2` and divide it by `x^2` to get `f'(x)=2/x`.

Explanation:

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

As it applies to the natural log function, the chain rule says:
`ln(u)'=u'*1/u=(u')/u`
Where `u` is a function of `x`.

In our case (`f(x)=ln(x^2)`), `u=x^2`, so the derivative is:
`f'(x)=(x^2')/(x^2)=(2x)/x^2=2/x`