Problem 31 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\frac{d^{2} x}{d y^{2}}\) equals (A) \(-\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3}\) (B) \(\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}\) (C) \(-\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}\left(\frac{d y}{d x}\right)^{-3}\) (D) \(\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-2}\)

Step-by-Step Solution

Answer

The answer is (A): -\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3}.

Step 1: Identify the relationship between derivatives

We are given the second derivative \( \frac{d^{2} x}{d y^{2}} \) and we need to express it in terms of \( \frac{d^{2} y}{d x^{2}} \). Begin by considering the relationship between \( x \) and \( y \). To find \( \frac{d^{2} x}{d y^{2}} \), we start with \( \frac{d y}{d x} \) and \( \frac{d x}{d y} = \left(\frac{d y}{d x}\right)^{-1} \).

Step 2: Use the chain rule for differentiation

By the chain rule, the derivative of \( x \) with respect to \( y \) is related to the derivative of \( y \) with respect to \( x \) as follows: if \( \frac{d y}{d x} = f'(x) \), then \( \frac{d x}{d y} = \left(\frac{d y}{d x}\right)^{-1} \).

Step 3: Differentiate again to find the second derivative

Differentiate \( \frac{d x}{d y} \) with respect to \( y \) to obtain the second derivative: \( \frac{d^{2} x}{d y^{2}} \). Use implicit differentiation on \( \frac{d x}{d y} = \left(\frac{d y}{d x}\right)^{-1} \), and set up the equation \( \frac{d}{dy}\left(\left(\frac{d y}{d x}\right)^{-1}\right) \).

Step 4: Substitute and simplify the expression using implicit differentiation

The differentiation step involves the derivative of the reciprocal, which results in: \( \frac{d}{dy}\left(\left(\frac{d y}{d x}\right)^{-1}\right) = -\left(\frac{d^{2} y}{d x^{2}}\right) \left(\frac{d x}{d y}\right)^{3} \). Substitute \( \frac{d x}{d y} = \left(\frac{d y}{d x}\right)^{-1} \) into the expression to get: \( -\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3} \).

Step 5: Match with the given options

Compare your derived expression with the given options to find the correct match. The expression \( -\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3} \) corresponds to option (A).

Key Concepts

Chain Rule

Implicit Differentiation and the Chain Rule are connected through the need to find derivatives of composed functions. The Chain Rule states that if you have a function inside another function, you can differentiate them by multiplying the derivative of the outer function by the derivative of the inner function. This rule is essential when differentiating complex functions.
For instance, if you have a composite function such as \( g(f(x)) \), where \( g \) and \( f \) are differentiable functions, the chain rule formula is \( (g \circ f)'(x) = g'(f(x)) \cdot f'(x) \).
This becomes quite handy in problems involving implicit differentiation. When equations are given implicitly, such as \( x \) and \( y \) are related by a complex function, we might find derivatives like \( \frac{dy}{dx} \) through the use of the Chain Rule. Hence, using the reciprocal \( \frac{dx}{dy} \) as the inverse of \( \frac{dy}{dx} \) is a pivotal step that sets the foundation for further derivations, especially when tackling second derivatives.

Second Derivative

The second derivative, denoted as \( \frac{d^2 y}{dx^2} \) or \( \frac{d^2 x}{dy^2} \) in the context of implicit differentiation, is simply the derivative of the first derivative. It helps us understand how the rate of change itself is changing.
Finding the second derivative involves two differentiation steps: calculating the first derivative and then differentiating it again.
When dealing with implicit differentiation, finding the second derivative can be slightly more complex, as it often involves applying the Chain Rule or other differentiation rules like the Reciprocal Rule iteratively.
In the problem provided, once you have \( \frac{dx}{dy} \), obtaining the second derivative \( \frac{d^2 x}{dy^2} \) requires differentiating \( \frac{dx}{dy} \) again with respect to \( y \), often applying the Chain Rule again implicitly.

Reciprocal Rule

The Reciprocal Rule is a useful tool for differentiating reciprocal expressions. If you have a function \( y = f(x) \), its reciprocal function is \( x = f^{-1}(y) \). Differentiating the reciprocal is straightforward when understood with respect to implicit differentiation.
The derivative of a reciprocal \( \frac{1}{f(x)} \) can be found using the formula: \( \frac{d}{dx} \left(\frac{1}{f(x)}\right) = -\frac{f'(x)}{(f(x))^2} \).
In implicit differentiation, once \( \frac{dy}{dx} \) is found, its reciprocal is \( \frac{dx}{dy} = (\frac{dy}{dx})^{-1} \).
For a second derivative, as illustrated in the exercise, the reciprocal's derivative is applied again to find \( \frac{d^2 x}{d y^2} \), often leading to expressions that might seem complex but reduce intuitively by substituting known derivatives. This understanding simplifies complex implicit differentiation tasks.