A) \[\frac{y}{x}\]
B) \[\frac{x+y}{xy}\]
C) \[xy\]
D) \[\frac{x}{y}\]
Correct Answer: A
Solution :
[a] \[{{x}^{m}}{{y}^{n}}={{(x+y)}^{m+n}}\] \[\therefore m\log x+n\log y=(m+n)In(x+y)\] Diffrentiating w.r.t.x, \[\therefore \frac{m}{x}+\frac{n}{y}\frac{dy}{dx}=\frac{m+n}{x+y}\left( 1+\frac{dy}{dx} \right)\] \[\Rightarrow \left( \frac{m}{x}-\frac{m+n}{x+y} \right)=\left( \frac{m+n}{x+y}-\frac{n}{y} \right)\frac{dy}{dx}\] \[\Rightarrow \frac{my-nx}{x(x+y)}=\left( \frac{my-nx}{y(x+y)} \right)\frac{dy}{dx}\Rightarrow \frac{dy}{dx}=\frac{y}{x}\]
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