Animated Solution for Mathematics - Differentiation: If $x=2\sin \theta -\sin 2\theta$ and $y=2\cos \theta -\cos 2\theta$, $\theta \in [0,2\pi ]$, then $\frac{d^{2}y}{d{x}^2}$ at $\theta =\pi$ is:

Welcome, fellow traveler of the JEE path. Today, we are exploring the elegant geometry of a parametric curve defined by:

Our mission is to find the second derivative $\frac{d^{2}y}{d{x}^2}$ at $\theta =\pi$. This is a classic test of your ability to handle parametric differentiation with grace and precision.

Before we can find the second derivative, we must find the first derivative, $\frac{dy}{dx}$. Since both $x$ and $y$ are functions of $\theta$, we use the parametric chain rule:

Differentiating $x$ and $y$ with respect to $\theta$:

Assembling these, we obtain:

Do not differentiate this quotient yet; it is a trap. Instead, we use the sum-to-product identities to simplify the expression.

The numerator $\sin 2\theta -\sin \theta$ becomes:

The denominator $\cos \theta -\cos 2\theta$ becomes:

The terms $2$ and $\sin \left(\frac{\theta }{2}\right)$ cancel out completely. We are left with the elegant result:

We need to find $\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)$. Since $\frac{dy}{dx}$ is a function of $\theta$, we apply the chain rule:

Differentiating $\cot \left(\frac{3\theta }{2}\right)$ with respect to $\theta$ gives:

Now, we multiply by $\frac{d\theta }{dx}$, which is the reciprocal of $\frac{dx}{d\theta }$:

Finally, we evaluate the expression at $\theta =\pi$. We calculate the trigonometric components:

Putting it all together:

We have arrived at the finish line. The final result is $\frac{3}{8}$. The positive result indicates that the curve is concave up at this point.