Question

Differentiate the following functions from first principles:

e^−x

Answer 1

$$\begin{gathered} \mathrm{Let}f\left(x\right)=e^{-x} \\ \Rightarrow f\left(x+h\right)=e^{-\left(x+h\right)} \\ \frac{d}{dx}\left\{f\left(x\right)\right\}=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{e^{-\left(x+h\right)}-e^{-x}}{h} \\ =\underset{h\to 0}{\lim }\frac{e^{-x}\times e^{-h}-e^{-x}}{h} \\ =\underset{h\to 0}{\lim }{e}^{-x}\left\{\frac{\left(e^{-h}-1\right)}{-h}\right\}\times \left(-1\right) \\ =-e^{-x}\underset{h\to 0}{\lim }\left\{\frac{\left(e^{-h}-1\right)}{-h}\right\} \\ =-e^{-x}\left[\because \underset{h\to 0}{\lim }\frac{e^{-h}-1}{-h}=1\right] \\ \mathrm{So},\frac{d}{dx}\left(e^{-x}\right)=-e^{-x} \end{gathered}$$