need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.4 question 7

$\left(\frac{a y-x-y}{x+y-a x}\right)$

Hint:

Use chain rule and product rule

Given:

$(x+y)^{2}=2 a x y$

Solution:

Differentiate the given equation w.r.t x

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

$\frac{d}{d x}\left[x^{2}+2 x y+y^{2}\right]=2 a \frac{d}{d x}(x y) \quad\left[\because(a+b)^{2}=a^{2}+2 a b+b^{2}\right]$

$\frac{d\left(x^{2}\right)}{d x}+\frac{d(2 x y)}{d x}+\frac{d y^{2}}{d x}=2 a\left[x \frac{d y}{d x}+y \frac{d x}{d x}\right] \quad\left[\because \frac{d(u v)}{d x}=u \frac{d v}{d x}+v \frac{d u}{d x}\right]$

$2 x+2 \frac{d(x y)}{d x}+\frac{d y^{2}}{d y} \times \frac{d y}{d x}=2 a\left[x \frac{d y}{d x}+y\right]$

$2 x+2\left[x \frac{d y}{d x}+y \frac{d x}{d x}\right]+2 y \frac{d y}{d x}=2 a x \frac{d y}{d x}+2 a y \quad\left[\because \frac{d\left(x^{n}\right)}{d x}=n x^{n-1}\right]$

$2 x+2 x \frac{d y}{d x}+2 y+2 y \frac{d y}{d x}=2 a x \frac{d y}{d x}+2 a y$

$2 x \frac{d y}{d x}+2 y \frac{d y}{d x}-2 a x \frac{d y}{d x}=2 a y-2 x-2 y$

$\frac{d y}{d x}[2 x+2 y-2 a x]=2 a y-2 x-2 y$

$\frac{d y}{d x}=\frac{2 a y-2 x-2 y}{2 x+2 y-2 a x}$

$\frac{d y}{d x}=\frac{2(a y-x-y)}{2(x+y-a x)}$

$\frac{d y}{d x}=\frac{a y-x-y}{x+y-a x}$

Hence $\frac{d y}{d x}=\frac{a y-x-y}{x+y-a x}$ is the required answer