Differentiate the following functions with respect to x :

$\frac{1 + 3^{x}}{1 - 3^{x}}$

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Solution

We have,

$\frac{d}{d x} (\frac{1 + 3^{x}}{1 - 3^{x}})$

Using quotient rule,

$= \frac{(1 - 3^{x}) \frac{d}{d x} (1 + 3^{x}) - (1 + 3^{x}) \frac{d}{d x} (1 - 3^{x})}{(1 - 3^{x})^{2}}$

$= \frac{(1 - 3^{x}) 3^{x} l o g 3 + (1 + 3^{x}) 3^{x} l o g 3}{(1 - 3^{x})^{2}}$

$= \frac{3^{x} l o g 3 - 3^{x} \times 3^{x} l o g 3 + 3^{x} l o g 3 + 3^{x} \times 3^{x} l o g 3}{(1 - 3^{x})^{2}} = \frac{2 \times 3^{x} l o g 3}{(1 - 3^{x})^{2}}$

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