Animated Solution for Mathematics - Differentiation: $\frac{d^{2}x}{d{y}^2}$ equals

Welcome, fellow traveler on the path to JEE mastery. Today, we are going to dismantle a classic calculus trap—a problem that looks deceptively simple but hides a profound lesson about the nature of derivatives. We are tasked with finding the second derivative of an inverse function, specifically $\frac{d^{2}x}{d{y}^2}$.

Many students, in the heat of an exam, will look at the first derivative $\frac{dx}{dy}={\left(\frac{dy}{dx}\right)}^{-1}$ and instinctively try to differentiate both sides to get $\frac{d^{2}x}{d{y}^2}=-{\left(\frac{d^{2}y}{d{x}^2}\right)}^{-1}$. Stop! Take a breath. That is the siren song of a false intuition. Let us derive the truth together.

The Inverse Function Theorem

We begin with the bedrock of this problem: the Inverse Function Theorem. We know that if $y$ is a function of $x$, then $x$ is a function of $y$. Their derivatives are reciprocals of each other. We write this as:

This is our starting point. We have successfully expressed the first derivative of $x$ with respect to $y$ in terms of the derivative of $y$ with respect to $x$.

Now, we must find the second derivative. By definition, this is the derivative of the first derivative with respect to $y$:

The Chain Rule

Here is where the magic—and the danger—lies. We have an operator $\frac{d}{dy}$ acting on a function of $x$. We cannot simply apply the power rule to ${\left(\frac{dy}{dx}\right)}^{-1}$ because the variable of differentiation ($y$) does not match the variable of the function ($x$).

To fix this, we invoke the Chain Rule. We need to change the operator from $\frac{d}{dy}$ to $\frac{d}{dx}$. The Chain Rule provides the bridge:

This is the most critical step in the entire derivation. We are essentially saying: "To see how this changes with $y$, first see how it changes with $x$, then multiply by how $x$ changes with $y$." Applying this to our expression:

Differentiating the Power

Now, we focus on the term inside the bracket. We are differentiating ${\left(\frac{dy}{dx}\right)}^{-1}$ with respect to $x$. This is a straightforward application of the power rule, followed by the chain rule for the inner function $\frac{dy}{dx}$:

The derivative of $\frac{dy}{dx}$ with respect to $x$ is, by definition, the second derivative $\frac{d^{2}y}{d{x}^2}$. So, our expression becomes:

We are almost there. Let us bring it all together. We substitute this result back into our chain rule equation:

Recall from our first step that $\frac{dx}{dy}={\left(\frac{dy}{dx}\right)}^{-1}$. Let us substitute that in as well:

Now, look at the exponents of $\frac{dy}{dx}$. We have a power of $-2$ multiplied by a power of $-1$. When we multiply terms with the same base, we add the exponents: $-2+(-1)=-3$.

And there it is. The elegance of the result is undeniable. We started with a simple inverse and ended with a beautiful, compact expression that perfectly captures the relationship between the second derivatives.

This result is not just a formula to memorize; it is a testament to the power of the Chain Rule. Whenever you feel lost in a sea of variables, remember: the Chain Rule is your compass. Keep practicing, keep questioning, and you will master the language of the universe.