I'm currently taking Calculus. I'm pretty good with derivatives apart from when it comes to logarithmic differentiation etc.
Here is one I'm having problems with, if anyone could help that would be appreciated.
$$ f(x)=\ln (\ln (x) ) .$$
Can someone please explain the derivative of this? Thanks!
Best Answer
The basic formula for the derivative of $\ln x$ is $${d\over dx}\ln x={1\over x}.$$
Recall the chain rule: ${d\over dx}f\bigl(g(x)\bigr) =f'\bigl(g(x)\bigr)\cdot g'(x)$.
So, by the chain rule, with $f(x)=\ln x$ and $g(x)=\ln x$, $${d\over dx}\ln (\ln x)={1\over \ln x}\cdot (\ln x )'={1\over \ln x}\cdot {1\over x}= {1\over x\ln x}.$$
Don't tell anyone I told you this, but you can remember: "the derivative of $\ln$ of something is (1 over the something) times the derivative of the something".