Q252 / 270 · NDA 2020 · Mathematics

If $x^m y^n=a^{m+n}$, then what is $\frac{d y}{d x}$ equal to?

  • A
    $\frac{m y}{n x}$
  • Correct answer
    B
    $-\frac{m y}{n x}$
  • C
    $\frac{m x}{n y}$
  • D
    $-\frac{n y}{m x}$

Solution

$x^m y^n=a^{m+n}$ $\frac{x^m}{a^m}=\frac{a^n}{y^n}$ (i) Differentiating equation (i) w.r.t. $x$, we get $\frac{m \cdot x^{(m-1)}}{a^m}=\frac{a^n \cdot(-n)}{y^{n+1}} \cdot \frac{d y}{d x}$ $\frac{d y}{d x}=\frac{m \cdot y^{(n+1)} \cdot(x)^{(m-1)}}{-n \cdot a^m \cdot a^n}$ $=-\frac{m y}{n \cdot x} \cdot\left(\frac{x^m \cdot y^n}{a^{m+n}}\right)=-\frac{m y}{n x}(1)=-\frac{m y}{n x}$

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

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