Differentiate the following functions with respect to x: e^ +( )^… - Vidyadip

Question

Differentiate the following functions with respect to x:
$\text{e}^{\sin\text{x}}+(\tan\text{x})^\text{x}$

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Answer

Let $\text{y}=\text{e}^{\sin\text{x}}+(\tan\text{x})^\text{x}$
$\Rightarrow\text{y}=\text{e}^{\sin\text{x}}+\text{e}^{\log(\tan\text{x})^\text{x}}$
$\Rightarrow\text{y}=\text{e}^{\sin\text{x}}+\text{e}^{\text{x}\log(\tan\text{x})}$
Differentiating with resepect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\text{e}^{\sin\text{x}}\big)+\frac{\text{d}}{\text{dx}}\big\{\text{e}^{\text{x}\log(\tan\text{x})}\big\}$
$=\text{e}^{\sin\text{x}}\frac{\text{d}}{\text{dx}}(\sin\text{x})+\text{e}^{\text{x}\log(\tan\text{x})}\frac{\text{d}}{\text{dx}}(\text{x}\log\tan\text{x})$
$=\text{e}^{\sin\text{x}}(\cos\text{x})+\text{e}^{\log(\tan\text{x})^\text{x}}\Big[\text{x}\frac{\text{d}}{\text{dx}}(\log\tan\text{x})+\log\tan\text{x}\frac{\text{d}}{\text{dx}}(\text{x})\Big]$
$=\text{e}^{\sin\text{x}}(\cos\text{x})+(\tan\text{x})^\text{x}\Big[\frac{\text{x}}{\tan\text{x}}(\sec^2\text{x})+\log\tan\text{x}\Big]$
$=\text{e}^{\sin\text{x}}(\cos\text{x})+(\tan\text{x})^\text{x}\big[\text{x}\sec\text{x cosec x}+\log\tan\text{x}\big]$

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