Differentiate the function with respect to x : 2 (x^ 2 ) - Vidyadip

Question

Differentiate the function with respect to x : $2 \sqrt{\cot \left(x^{2}\right)}$

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Answer

Given function is: $2 \sqrt{\cot \left(x^{2}\right)}$
Let y = $2 \sqrt{\cot \left(x^{2}\right)}$
$\Rightarrow \frac{d y}{d x}=\frac{d}{d x}(2 \sqrt{\cot \left(x^{2}\right)})$
we know that
$\frac{d}{d x}(k f(x))$ = $k \frac{d}{d x} f(x), \frac{d}{d x}(\sqrt{f(x)})$ = $\frac{1 \ \ \ \ \ \ \ \ \ \ d}{2 \sqrt{f(x)} d x}(f(x))$
Applying both the formula, we get,
$\frac{dy}{dx}=2 \cdot \frac{1}{2 \sqrt{\cot \left(\mathrm{x}^{2}\right)}} \cdot \frac{\mathrm{d}}{\mathrm{d} \mathrm{x}} \cot \left(\mathrm{x}^{2}\right)$
Now, $\frac{d}{d x}(\cot x)=-cosec ^{2} x$
Therefore,
$\frac{dy}{dx}=\frac{1}{\sqrt{\cot \left(x^{2}\right)}} \cdot\left[-\ cosec ^{2}\left(x^{2}\right)\right] \cdot \frac{d}{d x}\left(x^{2}\right)$
= $\sqrt{\frac{\sin x^{2}}{\cos x^{2}}} \times\left(-\frac{1}{\sin ^{2}\left(x^{2}\right)}\right) \times(2 x)$
= $\frac{-2 x}{\sin x^{2} \sqrt{\sin x^{2} \cos x^{2}}}$
= $\frac{-2 x}{\sin x \sqrt[2]{\sin x^{2} \cos x^{2}}} \times \frac{\sqrt{2}}{\sqrt{2}}$
= $\frac{-2 \sqrt{2} x}{\sin x \sqrt[2]{2 \sin x^{2} \cos x^{2}}}$ [Using sin 2x = 2 sin x cos x]
= $\frac{-2 \sqrt{2} x}{\sin x^{2} \sqrt{\sin 2 x^{2}}}$

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