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  • explain solution RD Sharma class 12 chapter Differentiation exercise 10.2 question 28 maths

explain solution RD Sharma class 12 chapter Differentiation exercise 10.2 question 28 maths

Answers (1)

Answer: \frac{1}{\sqrt{x^{2}+1}}

Hint:  You must know the rules of solving derivative of logarithm function.

Given: \log \left(x+\sqrt{x^{2}+1}\right)

Solution:

Differentiate with respect to x

\frac{d y}{d x}=\frac{d}{d x}\left[\log \left(x+\sqrt{x^{2}+1}\right)\right]

\frac{d y}{d x}=\frac{1}{x+\sqrt{x^{2}+1}} \frac{d}{d x}\left(x+\left(x^{2}+1\right)^{\frac{1}{2}}\right)

\frac{d y}{d x}=\frac{1}{x+\sqrt{x^{2}+1}}\left[1+\frac{1}{2}\left(x^{2}+1\right)^{\frac{1}{2}-1} \frac{d}{d x}\left(x^{2}+1\right)\right]

\frac{d y}{d x}=\frac{1}{x+\sqrt{x^{2}+1}}\left[1+\frac{1}{2 \sqrt{x^{2}+1}} \times 2 x\right]

\begin{aligned} &\frac{d y}{d x}=\frac{1}{x+\sqrt{x^{2}+1}}\left[\frac{\sqrt{x^{2}+1}+x}{\sqrt{x^{2}+1}}\right] \\ &\frac{d y}{d x}=\frac{1}{\sqrt{x^{2}+1}} \end{aligned}

 

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