Question

Find the derivative of $e^2x$.

Answer 1

Compute the derivative:

Suppose that $f$ and $g$ are two functions, then the chain rule states that,

$$\begin{gathered} \frac{d}{dx}\left[f\left(g\right(x\left)\right)\right]=f\text{'}\left(g\right(x\left)\right)\times \frac{d}{dx}\left[g\right(x\left)\right] \\ \frac{d}{dx}\left[f\left(g\right(x\left)\right)\right]=f\text{'}\left(g\right(x\left)\right)\times [g\text{'}(x\left)\right] \end{gathered}$$

So we can find the derivative as

$\begin{array}{rcl}\frac{d}{dx}\left[e^{2x}\right]& =& \frac{d}{dx}\left(e^{2x}\right)\times \frac{d}{dx}\left(2x\right)\\ & =& 2e^{2x}\end{array}$

Hence, the required derivative is $2e^{2x}$.