# 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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#### Explanation

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#### Explanation:

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Sep 14, 2016

Answer is ${3}^{x} \cdot \log 3$

#### Explanation:

Let $y = {3}^{x}$

Taking log on both sides,

$\log y = x \log 3$.........(Logarithmic property)

Differentiating both sides and applying chain rule,

$\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = \log 3$..........(log 3 is constant and was multiplied to variable x)

Therefore,

$\frac{\mathrm{dy}}{\mathrm{dx}} = {3}^{x} \log 3$

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