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NDA 2020
Exam
NDA
Year
2020
Subject
Mathematics
Chapter
Differentiation

NDA 2020 · Mathematics · Differentiation

If x^m y^n=a^ m+n , then what is (d y)/(d x) equal to?

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

Options

  1. Amynx\frac{m y}{n x}
  2. B−mynx-\frac{m y}{n x}
  3. Cmxny\frac{m x}{n y}
  4. D−nymx-\frac{n y}{m x}

Correct answer

B. −mynx-\frac{m y}{n x}

Step-by-step solution

xmyn=am+nxmam=anynx^m y^n=a^{m+n}\frac{x^m}{a^m}=\frac{a^n}{y^n} (i) Differentiating equation (i) w.r.t. xx, we get m⋅x(m−1)am=an⋅(−n)yn+1⋅dydxdydx=m⋅y(n+1)⋅(x)(m−1)−n⋅am⋅an=−myn⋅x⋅(xm⋅ynam+n)=−mynx(1)=−mynx\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}
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