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JEE Main 2006
Exam
JEE Main
Year
2006
Subject
Mathematics
Chapter
Differentiation

JEE Main 2006 · Mathematics · Differentiation

If x^m · y^n=(x+y)^ m+n , then (d y)/(d x) is

If xm⋅yn=(x+y)m+nx^m \cdot y^n=(x+y)^{m+n}, then dydx\frac{d y}{d x} is

Options

  1. Ayx\frac{y}{x}
  2. Bx+yxy\frac{x+y}{x y}
  3. Cxy
  4. Dxy\frac{x}{y}

Correct answer

A. yx\frac{y}{x}

Step-by-step solution

xm⋅yn=(x+y)m+n⇒mln⁡x+nln⁡y=(m+n)ln⁡(x+y)x^m \cdot y^n=(x+y)^{m+n} \Rightarrow m \ln x+n \ln y=(m+n) \ln (x+y) ∴mx+nydydx=m+nx+y(1+dydx)⇒(mx−m+nx+y)=(m+nx+y−ny)dydx\therefore \frac{m}{x}+\frac{n}{y} \frac{d y}{d x}=\frac{m+n}{x+y}\left(1+\frac{d y}{d x}\right) \Rightarrow\left(\frac{m}{x}-\frac{m+n}{x+y}\right)=\left(\frac{m+n}{x+y}-\frac{n}{y}\right) \frac{d y}{d x} my−nxx(x+y)=(my−nxy(x+y))dydx⇒dydx=yx\quad \frac{m y-n x}{x(x+y)}=\left(\frac{m y-n x}{y(x+y)}\right) \frac{d y}{d x} \Rightarrow \frac{d y}{d x}=\frac{y}{x}
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