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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Differentiation

NDA 2017 · Mathematics · Differentiation

What is (d^2 x)/(d y^2) equal to?

What is \frac{d^2 x}{d y^2}$ equal to?

Options

  1. A−-\left(\frac{d^2 y}{d x^2}\right)^{-1}(dydx)−3\left(\frac{d y}{d x}\right)^{-3}
  2. B\left(\frac{d^2 y}{d x^2}\right)^{-1}(dydx)−2\left(\frac{d y}{d x}\right)^{-2}
  3. C−-\left(\frac{d^2 y}{d x^2}\right)(dydx)−3\left(\frac{d y}{d x}\right)^{-3}
  4. D(d2ydx2)−1\left(\frac{d^2 y}{d x^2}\right)^{-1}

Correct answer

C. −-\left(\frac{d^2 y}{d x^2}\right)(dydx)−3\left(\frac{d y}{d x}\right)^{-3}

Step-by-step solution

Let y=f(x)⇒dydx=f1(x)⇒dxdy=1\mathrmf′(x)\mathrm{y}=\mathrm{f}(\mathrm{x}) \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{f}^1(\mathrm{x}) \Rightarrow \frac{\mathrm{dx}}{\mathrm{dy}}=\frac{1}{\mathrmf^{\prime}(\mathrm{x})}d2xdy2=−ddyf′(x)(f′(x))2...(i)−ddyf′(x)(f′(x))2=−ddy(dydx)⋅dxdx(f′(x))2=−ddx(f′(x))⋅dxdy(dydx)2⇒−f′(x)⋅dxdy(dydx)2=−(d2ydx2)(dydx)3=−d2ydx2(dydx)−3\begin{aligned} & \frac{d^2 x}{d y^2}=\frac{\frac{-d}{d y} f^{\prime}(x)}{\left(f^{\prime}(x)\right)^2}...(i) \\ & \frac{\frac{-d}{d y} f^{\prime}(x)}{\left(f^{\prime}(x)\right)^2}=\frac{\frac{-d}{d y}\left(\frac{d y}{d x}\right) \cdot \frac{d x}{d x}}{\left(f^{\prime}(x)\right)^2}=\frac{\frac{-d}{d x}\left(f^{\prime}(x)\right) \cdot \frac{d x}{d y}}{\left(\frac{d y}{d x}\right)^2} \\ & \Rightarrow \frac{-f^{\prime}(x) \cdot \frac{d x}{d y}}{\left(\frac{d y}{d x}\right)^2}=\frac{-\left(\frac{d^2 y}{d x^2}\right)}{\left(\frac{d y}{d x}\right)^3}=\frac{-d^2 y}{d x^2}\left(\frac{d y}{d x}\right)^{-3} \end{aligned}$$
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