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need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.2 question 46

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Answer: \frac{1}{\sqrt{x^{2}+4 x+1}}

Hint: you must know the rules of solving derivative of logarithm function

Given:   \log \left[x+2+\sqrt{x^{2}+4 x+1}\right]


Let  y=\log \left[x+2+\sqrt{x^{2}+4 x+1}\right]

Differentiate both side with respect to x

\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}} \log \left[x+2+\sqrt{x^{2}+4 x+1}\right]

=\frac{1}{\left[x+2+\sqrt{x^{4}+4 x+1}\right]} \times\left[1+0+\frac{1}{2}\left(x^{2}+4 x+1\right)^{\frac{-1}{2}} \frac{\mathrm{d}}{\mathrm{dx}}\left(x^{2}+4 x+1\right)\right]

\Rightarrow \frac{1+\frac{2 x+4}{2\left(\sqrt{x^{2}+4 x+1}\right)}}{\left[x+2+\sqrt{x^{2}+4 x+1}\right]}

=\frac{\sqrt{x^{2}+4 x+1}+x+2}{\left[x+2+\sqrt{x^{2}+4 x+1}\right] \times \sqrt{x^{2}+4 x+1}}

\Rightarrow \frac{1}{\sqrt{x^{2}+4 x+1}}


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