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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.
$\frac{\text{x}^{2}\cos\frac{\pi}{4}}{\sin\text{x}}$
✓

Answer

$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{x}^{2}\cos\frac{\pi}{4}}{\sin\text{x}}\Big)$
$=\cos\frac{\pi}{4}.\frac{\text{d}}{\text{dx}}\Big(\frac{\text{x}^{2}}{\sin\text{x}}\Big)$
$=\frac{\frac{1}{\sqrt{2}}\Big[\sin\text{x}.\frac{\text{d}}{\text{dx}}(\text{x})^{2}-\text{x}^{2}.\frac{\text{d}}{\text{dx}}(\sin\text{x})\Big]}{\sin^{2}\text{x}}$
$=\frac{1}{\sqrt{2}}\Big[\frac{\sin\text{x}2\text{x}-\text{x}^{2}\cos\text{x}}{\sin^{2}\text{x}}\Big]$
$=\frac{1}{\sqrt{2}}\Big[\frac{2\text{x}}{\sin\text{x}}-\frac{\text{x}^{2}\cos\text{x}}{\sin^{2}\text{x}}\Big]$
$=\frac{1}{\sqrt{2}}\big[2\text{x}\ \text{cosec}\text{x}-\text{x}^{2}\ \text{cot}\text{x}\ \text{cosec}\text{x}\big]$
$=\frac{\text{x}}{\sqrt{2}}\text{cosec}\text{x}\big[2-\text{x}\text{cot}\text{x}\big]$
Hence, the required answer is $\frac{\text{x}}{\sqrt{2}}\text{cosec}\text{x}\big[2-\text{x}\text{cot}\text{x}\big].$

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