# What is the derivative of 3^x?

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Jim G. Share
Sep 14, 2016

${3}^{x} \ln 3$

#### Explanation:

Begin by letting $y = {3}^{x}$

now take the ln of both sides.

$\ln y = \ln {3}^{x} \Rightarrow \ln y = x \ln 3$

differentiate $\textcolor{b l u e}{\text{implicitly with respect to x}}$

$\Rightarrow \frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = \ln 3$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = y \ln 3$

now y = ${3}^{x} \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = {3}^{x} \ln 3$

This result can be $\textcolor{b l u e}{\text{generalised}}$ as follows.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\frac{d}{\mathrm{dx}} \left({a}^{x}\right) = {a}^{x} \ln a} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

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Dec 11, 2017

$\frac{d}{\mathrm{dx}} \left({3}^{x}\right) = \ln \left(3\right) {3}^{x}$

#### Explanation:

$\frac{d}{\mathrm{dx}} \left({3}^{x}\right)$
$= \frac{d}{\mathrm{dx}} \left({e}^{x \ln \left(3\right)}\right)$ (since ${a}^{b} = {e}^{b \ln \left(a\right)}$)
$= \frac{d}{\mathrm{dx}} \left(x \ln \left(3\right)\right) \frac{d}{d} \left(x \ln \left(3\right)\right) \left({e}^{x \ln \left(3\right)}\right)$ (the chain rule)
$= \ln \left(3\right) {e}^{x \ln \left(3\right)}$
$= \ln \left(3\right) {3}^{x}$

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