Differentiation using definition of derivatives

Question

Use the definition of the derivatives to differentiate $f(x) = \ln x$.

Hint: Use the fact that $\displaystyle \lim_{t \to 0} (1+t)^{1/t} = e$.

Answer 1

Using the hint, we get: $$\lim_{t\rightarrow 0}\frac{\ln(1+t)}{t}=1$$

Then we use the definition of the derivative:

$$\begin{align*} \frac{d}{dx}\ln(x) &= \lim_{h\rightarrow 0}\frac{\ln(x+h)-\ln(x)}{h}\\ &=\lim_{h\rightarrow 0}\frac{\ln(\frac{x+h}{x}{})}{h}\\ &=\lim_{h\rightarrow 0}\frac{\ln(1+\frac{h}{x}{})}{x\cdot\frac{h}{x}} \end{align*}$$ Use our hint equation: $$=\lim_{h\rightarrow 0}\frac{1}{x}=\frac{1}{x}$$