Question

Differentiate the following functions from first principles:

e^3x

Answer 1

$$\begin{gathered} \mathrm{Let}f\left(x\right)=e^{3x} \\ \Rightarrow f\left(x+h\right)=e^{3\left(x+h\right)} \\ \frac{d}{dx}\left(f\left(x\right)\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{e^{3\left(x+h\right)}-e^{3x}}{h} \\ =\underset{h\to 0}{\lim }\frac{e^{3x}{e}^{3h}-e^{3x}}{h} \\ =\underset{h\to 0}{\lim }{e}^{3x}\left\{\frac{\left(e^{3h}-1\right)}{3h}\right\}\times 3 \\ =3e^{3x}\underset{h\to 0}{\lim }\left\{\frac{\left(e^{3h}-1\right)}{3h}\right\} \\ =3e^{3x} \\ \mathrm{Hence},\frac{d}{dx}\left(e^{3x}\right)=3e^{3x} \end{gathered}$$