Question

What is the derivative of $2^x$?

Answer 1

Determine the derivative of $2^x$.

Let $y=2^x$

Taking the natural logs on both sides of the above equation so we get:

$$\begin{gathered} \log y=\log \left(2^x\right) \\ \Rightarrow \log y=x\log \left(2\right)[\because \log a^m=m\log a] \end{gathered}$$

Differentiating both sides with respect to x, we get:

$$\begin{gathered} \frac{d\left(\log y\right)}{dx}=\log 2\frac{d\left(x\right)}{dx} \\ \Rightarrow \frac{1}{y}\frac{dy}{dx}=\log 2 \\ \Rightarrow \frac{dy}{dx}=y\log 2\left(\because y=2^x\right) \\ \Rightarrow \frac{dy}{dx}=2^x.\log 2 \end{gathered}$$

Hence, the derivative of $2^x$ is$"2^x.\log 2"$.