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provide solution for RD Sharma maths class 12 chapter Differentiation exercise  10.2 question 41

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Answer: \frac{3}{3 x+2}-\frac{2 x^{2}}{(2 x-1)}-2 x \log (2 x-1)

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

Given: \log (3 x+2)-x^{2} \log (2 x-1)

Solution:

Let  y=\log (3 x+2)-x^{2} \log (2 x-1)

Differentiate with respect to x,

\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\log (3 x+2)-\mathrm{x}^{2} \log (2 x-1)\right]

\frac{d y}{d x}=\frac{d}{d x} \log (3 x+2)-\frac{d}{d x}\left\{x^{2} \log (2 x-1)\right\}

\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{(3 x+2)} \frac{\mathrm{d}}{\mathrm{dx}}(3 x+2)-\left\{\mathrm{x}^{2} \frac{\mathrm{d}}{\mathrm{dx}} \log (2 x-1)+\log (2 x-1) \frac{\mathrm{d}}{\mathrm{dx}} \mathrm{x}^{2}\right\}

\frac{d y}{d x}=: \frac{3}{3 x+2}-\frac{2 x^{2}}{(2 x-1)}-2 x \log (2 x-1)

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