Question

Differentiate the following functions using first principle:

$f\left(x\right)=\ln x$

• A. $\frac{1}{x}$
• B. $\frac{-1}{x}$
• C. $e^x$
• D. $e^{-x}$

Answer 1

Answer: (A)

$\frac{1}{x}$

$f\left(x\right)=\ln x$
$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{lim}\frac{\ln (x+h)-\ln x}{h}$
$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{\ln (1+\frac{h}{x})}{h}=\frac{1}{x}\underset{h\to 0}{lim}\frac{\ln (1+\frac{h}{x})}{\left(\frac{h}{x}\right)}=\frac{1}{x}$