What is the derivative of $3^{x}$ $3^x$ ?

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Answer 1 · 2016-01-04T08:50:14

$\frac{d}{d x} 3^{x} = 3^{x} ln (3)$ $d/dx 3^x = 3^xln(3)$

An easy way of doing this is by using logarithmic differentiation.

To do this, we will use the following:

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

$\Rightarrow ln (y) = ln (3^{x}) = x ln (3)$ $=> ln(y) = ln(3^x) = xln(3)$

$\Rightarrow \frac{d}{d x} ln (y) = \frac{d}{d x} x ln (3)$ $=> d/dxln(y) = d/dxxln(3)$

$\Rightarrow \frac{1}{y} \frac{d y}{d x} = ln (3)$ $=> 1/y dy/dx = ln(3)$

$\Rightarrow \frac{d y}{d x} = y ln (3) = 3^{x} ln (3)$ $=> dy/dx = yln(3) = 3^xln(3)$

$:. d/dx 3^x = 3^xln(3)$

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