5. continuity and differentiation — Maths STD 12 Science — Question

$\Rightarrow m \ln x+n \ln y=(m+n) \ln (x+y)$

Differentiating both sides.

$\therefore \frac{m}{x}+\frac{n}{y} \frac{d y}{d x}=\frac{m+n}{x+y}\left(1+\frac{d y}{d x}\right)$

$\Rightarrow\left(\frac{m}{x}-\frac{m+n}{x+y}\right)=\left(\frac{m+n}{x+y}-\frac{n}{y}\right) \frac{d y}{d x}$

$\Rightarrow \frac{m y-n x}{x(x+y)}=\left(\frac{m y-n x}{y(x+y)}\right) \frac{d y}{d x}$

$\Rightarrow \frac{d y}{d x}=\frac{y}{x}$