Question
Differentiate the following functions with respect to x:$\frac{\text{e}^\text{x}}{1+\text{x}^2}$
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Answer
$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{e}^\text{x}}{1+\text{x}^2}\Big)$$=\frac{(1+\text{x}^2)\frac{\text{d}}{\text{dx}}(\text{e}^\text{x})-(\text{e}^\text{x})\frac{\text{d}}{\text{dx}}(1+\text{x}^2)}{(1+\text{x}^2)^2}$
$=\frac{(1+\text{x}^2)\text{e}^\text{x}-\text{e}^\text{x}\times\text{2x}}{(1+\text{x}^2)^2}$
$=\frac{\text{e}^\text{x}(1+\text{x}^2-\text{2x})}{(1+\text{x}^2)^2}$
$=\frac{\text{e}^\text{x}(1-\text{x})^2}{(1+\text{x}^2)^2}$
$=\frac{(1+\text{x}^2)\text{e}^\text{x}-\text{e}^\text{x}\times\text{2x}}{(1+\text{x}^2)^2}$
$=\frac{\text{e}^\text{x}(1+\text{x}^2-\text{2x})}{(1+\text{x}^2)^2}$
$=\frac{\text{e}^\text{x}(1-\text{x})^2}{(1+\text{x}^2)^2}$
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