Question

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

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

Answer 1

Answer: (D)

$\because x^m\cdot y^m={\left(x+y\right)}^{m+n}$
Taking log on both sides, we get
$mlogx+nlogy=\left(m+n\right)log\left(x+y\right)$
On differentiating with respect to $x$, we get
$\frac{m}{x}+\frac{n}{y}\frac{dy}{dx}=\frac{\left(m+n\right)}{\left(x+y\right)}\left(1+\frac{dy}{dx}\right)$
$\Rightarrow \frac{dy}{dx}\left(\frac{m+n}{x+y}-\frac{n}{y}\right)=\frac{m}{x}-\frac{m+n}{x+y}$
$\Rightarrow \frac{dy}{dx}\left(\frac{my+ny-nx-ny}{y\left(x+y\right)}\right)$
$=\frac{mx+my-mx-nx}{x\left(x+y\right)}$
$\Rightarrow \frac{dy}{dx}=\frac{y}{x}$