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Proof of the Derivative of $\ln(x)$

Find the derivative of $f\left(x\right)=\ln x$ from first principles (1):
$f'(x)=\lim \limits_{h \to 0}$

Write the denominator as $\displaystyle \frac{1}{h}$:
$\displaystyle f'(x)=\lim \limits_{h \to 0} \frac{1}{h} \Bigl($