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Limits and Derivatives — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSLimits and Derivatives3 Marks
Question
Differentiate the following functions.
$\sin^{3}\text{x}\cos^{3}\text{x}$
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Answer

$\frac{\text{d}}{\text{dx}}(\sin^{3}\text{x}\cos^{3}\text{x})$
$=\sin^{3}\text{x}\cdot\frac{\text{d}}{\text{dx}}\cos^{3}\text{x}\cdot\frac{\text{d}}{\text{dx}}(\sin^{3}\text{x})$
$=\sin^{3}\text{x}.3\cos^{3}\text{x}(-\sin\text{x})+\cos^{3}\text{x}\cdot3\sin^{2}\text{x}\cdot\cos\text{x}$
$=-3\sin^{4}\text{x}\cos^{2}+3\cos^{4}\text{x}\sin^{2}\text{x}$
$=3\sin^{2}\text{x}\cos^{2}\text{x}(-\sin^{2}\text{x}+\cos^{2}\text{x})$
$=3\sin^{2}\text{x}\cos^{2}\text{x}\cdot\cos2\text{x}$
$=\frac{3}{4}\cdot4\sin^{2}\text{x}\cos^{2}\text{x}\cdot\cos2\text{x}$
$=\frac{3}{4}(2\sin\text{x}\cos\text{x})^{2}\cos2\text{x}$
$=\frac{3}{4}\sin^{2}2\text{x}\cdot\cos2\text{x}$
Hence, the required answer is $\frac{3}{4}\sin^{2}2\text{x}\cdot\cos2\text{x}.$

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