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Provide Solution for RD Sharma Class 12 Chapter 11 Higher Order Derivatives Exercise Multiple Choice Question question 20

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We must know about the derivative of logarithm.


                y=\log _{e}\left(\frac{x}{a+b x}\right)^{x}


                y=\log _{e}\left(\frac{x}{a+b x}\right)^{x}

                \\ y=x \log _{e}\left(\frac{x}{a+b x}\right) \

                \begin{aligned} &\ &\frac{y}{x}=\log x-\log (a+b x) \end{aligned}

Differentiate with respect to x 

                \frac{x \frac{d y}{d x}-y}{x^{2}}=\frac{1}{x}-\frac{b}{a+b x} \

                \begin{aligned} &\ &x \frac{d y}{d x}-y=x-\frac{b x^{2}}{a+b x} \end{aligned}

Differentiate with respect to x

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

                x \frac{d^{2} y}{d x^{2}}=\left(\frac{(a+b x)^{2}-(a+b x) 2 b x-b^{2} x^{2}}{(a+b x)^{2}}\right) \\

                \begin{aligned} & &x \frac{d^{2} y}{d x^{2}}=\left(\frac{(a+b x)[a+b x-2 b x]+b^{2} x^{2}}{(a+b x)^{2}}\right) \end{aligned}

                x \frac{d^{2} y}{d x^{2}}=\left(\frac{(a+b x)(a-b x)+b^{2} x^{2}}{(a+b x)^{2}}\right)

                \begin{aligned} &\\ &x \frac{d^{2} y}{d x^{2}}=\left(\frac{a^{2}-b^{2} x^{2}+b^{2} x^{2}}{(a+b x)^{2}}\right) \end{aligned}

                x \frac{d^{2} y}{d x^{2}}=\left(\frac{a^{2}}{(a+b x)^{2}}\right)=\left(\frac{a}{a+b x}\right)^{2}

                \\ x \frac{d^{2} y}{d x^{2}}=\left(\frac{a}{a+b x}\right)^{2} \                                                                                                                                         … (i)

And        \begin{aligned} \ &y=x \log \left(\frac{x}{a+b x}\right) \end{aligned}

Differentiate with respect to  x

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

                \frac{d y}{d x}=\frac{a}{a+b x}+\frac{y}{x}

                \begin{aligned} \\ &\frac{d y}{d x}-\frac{y}{x}=\frac{a}{a+b x} \end{aligned}

From (i)

                x \frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}-\frac{y}{x}\right)^{2}

                \begin{aligned} & \\ &x^{2} \frac{d^{2} y}{d x^{2}}=\frac{1}{x^{2}}\left(x \frac{d y}{d x}-y\right)^{2} \end{aligned}

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

                \begin{aligned} &\\ &x^{3} y_{2}=\left(x y_{1}-y\right)^{2} \end{aligned}

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