#### Provide solution for RD Sharma maths class 12 chapter 10 Differentiation exercise Multiple choice question 27

$\frac{x^{2}}{y^{2}} \sqrt{\frac{1-y^{6}}{1-x^{6}}}$

Hint:

Differentiate the function w.r.t x

Given:

$\sqrt{1-x^{6}}+\sqrt{1-y^{6}}=a^{3}\left(x^{3}-y^{3}\right)$

Solution:

We have, $\sqrt{1-x^{6}}+\sqrt{1-y^{6}}=a^{3}\left(x^{3}-y^{3}\right)$

Putting $x^{3}=\sin A, y^{3}=\sin B$

\begin{aligned} &\sqrt{1-\sin ^{2} A}+\sqrt{1-\sin ^{2} B}=a(\sin A-\sin B) \\\\ &\cos A+\cos B=a(\sin A-\sin B) \end{aligned}

\begin{aligned} &2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)=2 a \sin \left(\frac{A-B}{2}\right) \cos \left(\frac{A+B}{2}\right) \\\\ &\cot \left(\frac{A-B}{2}\right)=a^{3} \end{aligned}

\begin{aligned} &\frac{A-B}{2}=\cot ^{-1}\left(a^{3}\right) \\\\ &A-B=2 \cot ^{-1}\left(a^{3}\right) \\\\ &\sin ^{-1} x^{3}-\sin ^{-1} y^{3}=2 \cot ^{-1}\left(a^{3}\right) \end{aligned}

\begin{aligned} &\frac{1}{\sqrt{1-x^{6}}} \times \frac{d}{d x}\left(x^{3}\right)-\frac{1}{\sqrt{1-y^{6}}} \times \frac{d}{d x}\left(y^{3}\right)=0 \\\\ &\frac{1}{\sqrt{1-x^{6}}} \times 3 x^{2}-\frac{1}{\sqrt{1-y^{6}}} \times 3 y^{2} \times \frac{d y}{d x}=0 \end{aligned}

$\frac{d y}{d x}=\frac{x^{2}}{y^{2}} \sqrt{\frac{1-y^{6}}{1-x^{6}}}$