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explain solution RD Sharma class 12 chapter Differentiation exercise 10.2 question 51 maths

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Answer: \frac{1}{\sqrt{\cos x}}\left\{2 x+\frac{x^{2}}{2 \cos x}+\tan x\right\}

Hint: you must know the rule of solving derivative of trigonometric function

Given:   \frac{x^{2}+2}{\sqrt{\cos x}}


Let  y= \frac{x^{2}+2}{\sqrt{\cos x}}

Differentiate with respect to x

\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\sqrt{\cos x} \frac{\mathrm{d}}{\mathrm{dx}}\left(x^{2}+2\right)-\left(x^{2}+2\right) \frac{\mathrm{d}}{\mathrm{dx}}(\sqrt{\cos x})}{(\sqrt{\cos x})^{2}} \cdot \cdot \frac{d}{d x}\left(\frac{u}{v}\right)=\frac{v \frac{d u}{d x}-u \frac{d v}{d x}}{v^{2}}

    \begin{aligned} &=\frac{2 x \sqrt{\cos x}-\left(x^{2}+2\right)\left(\frac{1}{2} \frac{-\sin x}{\sqrt{\cos x}}\right)}{\cos x} \\\\ &\Rightarrow \frac{2 x \sqrt{\cos x}+\frac{\left(x^{2}+2\right) \sin x}{2 \sqrt{\cos x}}}{\cos x} \end{aligned}

    \begin{aligned} &\Rightarrow \frac{4 x \cos x+\left(x^{2}+2\right) \sin x}{2 \cos x^{\frac{3}{2}}} \\\\ &\Rightarrow \frac{2 x}{\sqrt{\cos x}}+\frac{1}{2} \frac{\left(x^{2}+2\right) \sin x}{(\cos x)^{\frac{3}{2}}} \end{aligned}

    \begin{aligned} &\Rightarrow \frac{1}{\sqrt{\cos x}}\left\{2 x+\frac{1}{2} \frac{\left(x^{2}+2\right) \sin x}{\cos x}\right\} \\\\ &\Rightarrow \frac{1}{\sqrt{\cos x}}\left\{2 x+\frac{x^{2}}{2 \cos x}+\tan x\right\} \end{aligned}

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