Differentiate the following function with respect to x:x^4(5 -3 ) - Vidyadip

Question

Differentiate the following function with respect to x:$\text{x}^4(5\sin\text{x}-3\cos\text{x})$

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Answer

Let $\text{u}=\text{x}^4;\text{v}=5\sin\text{x}-3\cos\text{x}$Then, $\text{u}'=\text{4x}^3;\text{v}'=5\cos\text{x}-3(-\sin\text{x})=5\cos\text{x}+3\sin\text{x}$
Using the product rule:
$\frac{\text{d}}{\text{dx}}(\text{uv})=\text{u}'\text{v}+\text{uv}'$
$\frac{\text{d}}{\text{dx}}(\text{x}^4(5\sin\text{x}-3\cos\text{x}))=\text{x}^4(5\cos\text{x}+3\sin\text{x})+\text{4x}^3(5\sin\text{x}-3\cos\text{x})$
$=\text{x}^3(\text{5x}\cos\text{x}+\text{3x}\sin\text{x}+20\sin\text{x}-12\cos\text{x})$
$=\text{x}^3((\text{3x}+20)\sin\text{x}+(\text{5x}-12)\cos\text{x})$
$=\text{3x}^4\sin\text{x}+20\text{x}^3\sin\text{x}+\text{5x}\cos\text{x}-12\cos\text{x}$

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