Q94 / 150 · JEE Main 2006 · Mathematics

If $x^m \cdot y^n=(x+y)^{m+n}$, then $\frac{d y}{d x}$ is

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

Solution

$x^m \cdot y^n=(x+y)^{m+n} \Rightarrow m \ln x+n \ln y=(m+n) \ln (x+y)$ $\therefore \frac{m}{x}+\frac{n}{y} \frac{d y}{d x}=\frac{m+n}{x+y}\left(1+\frac{d y}{d x}\right) \Rightarrow\left(\frac{m}{x}-\frac{m+n}{x+y}\right)=\left(\frac{m+n}{x+y}-\frac{n}{y}\right) \frac{d y}{d x}$ $\quad \frac{m y-n x}{x(x+y)}=\left(\frac{m y-n x}{y(x+y)}\right) \frac{d y}{d x} \Rightarrow \frac{d y}{d x}=\frac{y}{x}$

Hence, the correct answer is option: 'A'.

Relevant Questions