Question

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

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

Answer 1

Answer: (D)

$\frac{\text{y}}{\text{x}}$

Given $x^m{y}^n=(x+y{)}^{m+n}$

Taking log on both sides

$log\left(x^m{y}^n\right)=log(x+y{)}^{m+n}$

$mlogx+nlogy=m+nlog(x+y)$

Differentiating w.r.t. x

$\frac{m}{x}+\frac{n}{y}\frac{dy}{dx}=\frac{m+n}{x+y}[1+\frac{dy}{dx}]$

$\frac{m}{x}-\frac{m+n}{x+y}=(\frac{m+n}{x+y}-\frac{n}{y})\frac{dy}{dx}$

$\frac{mx+my-mx-nx}{x(x+y)}=\left(\frac{my+ny-nx-ny}{y(x+y)}\right)\frac{dy}{dx}$

$\Rightarrow \frac{dy}{dx}=\frac{y}{x}$.