Differentiate the following functions with respect to x: 3^x x+ - Vidyadip

Question

Differentiate the following functions with respect to x:$\frac{3^\text{x}}{\text{x}+\tan\text{x}}$

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Answer

We have,$\frac{\text{d}}{\text{dx}}\Big(\frac{3^\text{x}}{\text{x}+\tan\text{x}}\Big)$
Using quotient rule, we get
$\frac{(\text{x}+\tan\text{x})\frac{\text{d}}{\text{dx}}(3^\text{x})-(3^\text{x})\frac{\text{d}}{\text{dx}}(\text{x}+\tan\text{x})}{(\text{x}+\tan\text{x})^2}$
$=\frac{(\text{x}+\tan\text{x})\times3^\text{x}\log3-3^\text{x}(1+\sec^2\text{x})}{(\text{x}+\tan\text{x})^2}$
$=\frac{3^\text{x}\Big\{(\text{x}+\tan\text{x})\log3-(1+\sec^2\text{x})\Big\}}{(\text{x}+\tan\text{x})^2}$

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