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Please solve RD Sharma class 12 chapter Differentiation exercise 10.2 question 5 maths textbook solution

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Answer: : \frac{\cos \sqrt{x} e^{\sin \sqrt{x}}}{2 \sqrt{x}}

Hint: You must know the rules of solving derivation of exponential function and trigonometric function.

Given: e^{\sin \sqrt{x}}


Let y=e^{\sin \sqrt{x}}

Differentiating with respect to x,

\begin{aligned} &\frac{d y}{d x}=\frac{d}{d x}\left(e^{\sin \sqrt{x}}\right) \\ &\frac{d y}{d x}=e^{\sin \sqrt{x}} \frac{d}{d x}(\sin \sqrt{x}) \end{aligned}                [using chain rule]

\frac{d y}{d x}=e^{\sin \sqrt{x}} \times \cos \sqrt{x} \frac{d}{d x} \sqrt{x}            [again using chain rule]

\begin{aligned} &\frac{d y}{d x}=e^{\sin \sqrt{x}} \times \cos \sqrt{x} \times \frac{1}{2 \sqrt{x}} \\ &\frac{d y}{d x}=\frac{\cos \sqrt{x} e^{\sin \sqrt{x}}}{2 \sqrt{x}} \end{aligned}



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