Question

Find the derivative of log x by ab- intio method
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Answer 1

Answer:

The usual example where learning about the derivative is obtaining it for f(x)=x2 from first principles (see this for example).

I am stumped on how use first principles to obtain the derivative of a natural logarithm. We need:

limh→0ln(x+h)−lnxh=limh→0ln(1+hx)h

Now I am stuck. Of course I know that Taylor expansion around very small x of ln(1+x)=x−x22+x33−…, but that's not something one is supposed to know when learning first principles of differentiation. Is there something clever that I am missing?