Question

If $x^m.y^n=(x+y{)}^{m+n}$, then $\frac{dy}{dx}$ is

• A. $\frac{y}{x}$
• B. $\frac{x+y}{xy}$
• C. $xy$
• D. $\frac{x}{y}$

Answer 1

The correct option is A

$\frac{y}{x}$

$x^m.y^n=(x+y{)}^{m+n}$
$\Rightarrow mln\left(x\right)+nln\left(y\right)=(m+n)ln(x+y)$
Differentiating both sides.
$\therefore \frac{m}{x}+\frac{n}{y}\frac{dy}{dx}=\frac{m+n}{x+y}(1+\frac{dy}{dx})$
$\Rightarrow (\frac{m}{x}-\frac{m+n}{x+y})=(\frac{m+n}{x+y}-\frac{n}{y})\frac{dy}{dx}$
$\Rightarrow \frac{my-nx}{x(x+y)}=\left(\frac{my-nx}{y(x+y)}\right)\frac{dy}{dx}\Rightarrow \frac{dy}{dx}=\frac{y}{x}$