What is {d²x}{dy²} equal to?

Correct Answer is : −(d2ydx2)(dydx)−3-({d²y}{dx²})(dy/dx)^{-3}−(dx2d2y)(dxdy)−3

-({d²y}{dx²})^{-1}(dy/dx)^{-3}

({d²y}{dx²})^{-1}(dy/dx)^{-2}

Correct Answer is : −(d2ydx2)(dydx)−3-({d²y}{dx²})(dy/dx)^{-3}−(dx2d2y)(dxdy)−3

Test Index

UPSC NDA I 2017 Mathematics Solved Paper

Question : 66 of 120

Marks: +1, -0

What is

\frac{d^{2}x}{dy^{2}}

equal to?

Solution:

Concept:

The second derivative of

x

with respect to

y

is obtained using the inverse function relationship and the chain rule.

Explanation:

Start with the first derivative:

\frac{dx}{dy} = \left( \frac{dy}{dx} \right)^{-1}

Differentiate again with respect to

y

using the chain rule:

\frac{d^2x}{dy^2} = \frac{d}{dy} \left( \frac{dx}{dy} \right) = \frac{d}{dx} \left( \left( \frac{dy}{dx} \right)^{-1} \right) \cdot \frac{dx}{dy}

Differentiate

\left( \frac{dy}{dx} \right)^{-1}

with respect to

x

\frac{d}{dx} \left( \left( \frac{dy}{dx} \right)^{-1} \right) = - \left( \frac{dy}{dx} \right)^{-2} \cdot \frac{d^2y}{dx^2}

Multiply by

\frac{dx}{dy} = \left( \frac{dy}{dx} \right)^{-1}

\frac{d^2x}{dy^2} = - \left( \frac{dy}{dx} \right)^{-2} \cdot \frac{d^2y}{dx^2} \cdot \left( \frac{dy}{dx} \right)^{-1} = - \frac{d^2y}{dx^2} \left( \frac{dy}{dx} \right)^{-3}

Answer:

Option C:

- \left( \frac{d^2y}{dx^2} \right) \left( \frac{dy}{dx} \right)^{-3}

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