Question

The Derivative of $e^x$

• A. $e^x$
• B. $e^{x+1}$
• C. $x{e}^x$
• D. $e^{2x}$

Answer 1

The correct option is A

$e^x$

To find derivate of $e^x$.
$\underset{h\to 0}{lim}=\frac{e^{(x+h)-e^x}}{h}=\underset{h\to 0}{lim}\frac{e^x.e^h-e^x}{h}$
= $\underset{h\to 0}{lim}\frac{e^x(e^h-1)}{h}$
Expansion of $e^h=1+h+\frac{1}{2}{h}^2+\frac{1}{6}{h}^3+.....$
$\therefore \left(\frac{e^h-1}{h}\right)=\frac{(1+h+\frac{1}{2}{h}^2+....)-1}{h}$
= $\frac{h}{h}+\frac{1}{2}h+\frac{1}{6}{h}^2+....$
as $h\to 0=\left(\frac{e^h-1}{h}\right)\to 1+0+0+....$
$\therefore \underset{h\to 0}{lim}\frac{e^x(e^h-1)}{h}=e^x\left(1\right)=e^x$
$\therefore f^{\prime }\left(x\right)=e^x\text{where}f\left(x\right)=e^x$.