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

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

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What have you tried? Where did you get stuck? – fgp Oct 11 '12 at 22:21
By providing the hint, are you implying you know the answer (and want to see if others can figure it out), or is it (rather) that you're working on homework or doing self-learning? – Firefeather Oct 11 '12 at 22:28

1 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: $$=\lim_{h\rightarrow 0}\frac{1}{x}=\frac{1}{x}$$

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