Differentiate the following function with respect to x: x^2 4 - Vidyadip

Question

Differentiate the following function with respect to x:$\frac{\text{x}^2\cos\frac{\pi}{4}}{\sin\text{x}}$

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Answer

$\frac{\text{x}^2\cos\frac{\pi}{4}}{\sin\text{x}}=\text{x}^2\cos\frac{\pi}{4}\text{cosec}\text{x}$Let $\text{u}=\text{x}^2;\text{v}=\cos\frac{\pi}{4};\text{w}=\text{cosec}\text{x}$
Then, $\text{u}'=\text{2x};\text{v}'=0;\text{w}'=-\text{cosecx}\cot\text{x}$
Using the product rule:
$\frac{\text{d}}{\text{dx}}(\text{uvw})=\text{u}'\text{vw}+\text{uv}'\text{w}+\text{uv}\text{w}'$
$\frac{\text{d}}{\text{dx}}(\text{x}^2\cos\frac{\pi}{4}\text{cosec}\text{x})=\text{2x}\cos\frac{\pi}{4}\text{cosec}\text{x}+\text{x}^2.0.\text{\cosecx}+\text{x}^2\cos\frac{\pi}{4}(-\text{cosecx}\cot\text{x})$
$=\cos\frac{\pi}{4}(\text{2x}\text{cosecx}-\text{x}^2\text{cosecx}\cot\text{x})$
$=\cos\frac{\pi}{4}\Big(\frac{\text{2x}}{\sin\text{x}}-\text{x}^2\frac{\cot\text{x}}{\sin\text{x}}\Big)$

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