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