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

Question

Differentiate the following function with respect to x:$(\text{x}\sin\text{x}+\cos\text{x})(\text{e}^\text{x}+\text{x}^2\log\text{x})$

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Answer

$\text{u}=(\text{x}\sin\text{x}+\cos\text{x});\text{v}=(\text{e}^\text{x}+\text{x}^2\log\text{x})$$\text{u}'=\text{x}\cos\text{x}+\sin\text{x}-\sin\text{x}=\text{x}\cos\text{x}$
$\text{v}'=\text{e}^\text{x}+\text{x}+\text{2x}\log\text{x}$
Using the product rule:
$\frac{\text{d}}{\text{dx}}(\text{uv})=\text{uv}'+\text{vu}'$
$\frac{\text{d}}{\text{dx}}=[(\text{x}\sin\text{x}+\cos\text{x})(\text{e}^\text{x}+\text{x}^2\cos\text{x})]$
$=(\text{x}\sin\text{x}+\cos\text{x})(\text{e}^\text{x}+\text{x}+\text{2x}\log\text{x})+(\text{e}^\text{x}+\text{x}^2\log\text{x})(\text{x}\cos\text{x})$

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