The usual example where learning about the derivative is obtaining it for $f(x)=x^2$ 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:
$$\lim_{h\rightarrow0}\frac{\ln(x+h)-\ln x}{h}=\lim_{h\rightarrow0}\frac{\ln(1+\frac{h}{x})}{h}$$
Now I am stuck. Of course I know that Taylor expansion around very small $x$ of $\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\ldots$, but that's not something one is supposed to know when learning first principles of differentiation. Is there something clever that I am missing?
Best Answer
You should know that $$\lim_{k \to 0} \frac{\log(1+k)}{k}=1$$ Then, calling $k= \frac hx$, you get $$\lim_{h \to 0} \frac{\log(1+\frac hx)}{h}= \lim_{h \to 0} \frac{\log(1+\frac hx)}{x\frac hx} = 1 \cdot \frac{1}{x}= \frac{1}{x}$$