Differentiate the following function with respect to x:x^3e^x - Vidyadip

Question

Differentiate the following function with respect to x:$\text{x}^3\text{e}^\text{x}\cos\text{x}$

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Answer

Let $\text{u}=\text{x}^3;\text{v}=\text{e}^\text{x};\text{w}=\cos\text{x}$Then, $\text{u}'=\text{3x}^2;\text{v}'=\text{e}^\text{x};\text{w}'=-\sin\text{x}$
Using the product rule:
$\frac{\text{d}}{\text{dx}}(\text{uvw})=\text{u}'\text{vw}+\text{uv}'\text{w}+\text{uv}\text{w}'$
$\frac{\text{d}}{\text{dx}}(\text{x}^3\text{e}^\text{x}\cos\text{x})=\text{3x}^2\text{e}^\text{x}\cos\text{x}+\text{x}^3\text{e}^\text{x}\cos\text{x}+\text{x}^3\text{e}^\text{x}(-\sin\text{x})$
$=\text{x}^2\text{e}^\text{x}(\text{3}\cos\text{x}+\text{x}\cos\text{x}-\text{x}\sin\text{x})$

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