Animated Solution for Mathematics - Differentiation: If $x^2+y^2+\sin y=4$, then the value of $\frac{d^{2}y}{d{x}^2}$ at the point (-2,0) is

Welcome, fellow traveler on the road to JEE Advanced. Today, we are going to peel back the layers of a problem that, at first glance, seems like a tangled mess of variables.

We are given the equation $x^2+y^2+\sin y=4$ and asked to find the second derivative, $\frac{d^{2}y}{d{x}^2}$, at the specific point $(-2,0)$.

Many students see the $\sin y$ and immediately freeze, wondering how to isolate $y$. But here is the secret: you don't have to.

In the world of implicit differentiation, we treat $y$ as a hidden function of $x$. We don't need to see the function to understand its rate of change; we just need to respect the rules of the calculus game.

The First Derivative

Imagine you are standing on the curve defined by $x^2+y^2+\sin y=4$. To find the slope, we differentiate both sides with respect to $x$.

Using the power rule and the chain rule, we get:

This yields $2x+2y\frac{dy}{dx}+\cos y\frac{dy}{dx}=0$. Notice how the $\frac{dy}{dx}$ appears naturally from the chain rule.

Now, let's factor it out: $2x+(2y+\cos y)\frac{dy}{dx}=0$.

At the point $(-2,0)$, we substitute $x=-2$ and $y=0$. The equation becomes $2(-2)+(2(0)+\cos (0))\frac{dy}{dx}=0$.

Since $\cos (0)=1$, this simplifies beautifully to $-4+\frac{dy}{dx}=0$, which gives us our first derivative: $\frac{dy}{dx}=4$. We have successfully captured the slope at that point!

The Second Derivative

Now, the real challenge begins. We need the second derivative.

We take our equation $2x+(2y+\cos y)\frac{dy}{dx}=0$ and differentiate it again with respect to $x$. This is where the Product Rule is your best friend.

Let $u=(2y+\cos y)$ and $v=\frac{dy}{dx}$. The derivative of $u$ is $(2-\sin y)\frac{dy}{dx}$.

Applying the product rule, we get:

Substituting our derivative of $u$, we arrive at:

The Grand Finale

This looks like a mountain of algebra, but look at what happens when we plug in our known values: $x=-2$, $y=0$, and $\frac{dy}{dx}=4$.

Since $\sin (0)=0$ and $\cos (0)=1$, the equation collapses into:

Therefore, the final result is:

See how the complexity vanished? The beauty of mathematics lies in this exact moment—where the initial chaos resolves into a single, elegant number.

You didn't need to solve for $y$; you just needed to trust the process of differentiation. Keep this confidence with you as you tackle your next problem. You are capable of solving anything.