NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

With the usual notation, $\frac{d^2x}{dy^2}$ is:

Choose the correct answer:

Explanation

1. First Derivative Relation:

By the rule of inverse functions:

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

2. Differentiating with respect to $y$ :

To find $\frac{d^2x}{dy^2}$ , we differentiate the first derivative with respect to $y$ :

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

3. Applying the Chain Rule:

Since the term inside is a function of $x$ , we must use the chain rule $\frac{d}{dy} = \frac{dx}{dy} \cdot \frac{d}{dx}$ :

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

4. Performing the Differentiation:

Differentiate $\left( \frac{dy}{dx} \right)^{-1}$ with respect to $x$ using the power rule:

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

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

5. Substituting back:

Now, multiply this by $\frac{dx}{dy}$ (which is $\left( \frac{dy}{dx} \right)^{-1}$ ):

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

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

Correct Option:

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