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Question: If \(x^{m}y^{n} = 2(x + y)^{m + n},\) the value of \(\frac{dy}{dx}\) is...

If xmyn=2(x+y)m+n,x^{m}y^{n} = 2(x + y)^{m + n}, the value of dydx\frac{dy}{dx} is

A

x+yx + y

B

xy\frac{x}{y}

C

yx\frac{y}{x}

D

x−yx - y

Answer

yx\frac{y}{x}

Explanation

Solution

xmyn=2(x+y)m+nx^{m}y^{n} = 2(x + y)^{m + n}

⇒mlog⁡x+nlog⁡y=log⁡2+(m+n)log⁡(x+y)m\log x + n\log y = \log 2 + (m + n)\log(x + y) Differentiating w.r.t. x both sides

mx+nydydx=m+nx+y[1+dydx]\frac{m}{x} + \frac{n}{y}\frac{dy}{dx} = \frac{m + n}{x + y}\left\lbrack 1 + \frac{dy}{dx} \right\rbrack⇒ dydx=yx\frac{dy}{dx} = \frac{y}{x}.