Animated Solution for Mathematics - Differentiation: Let $y=f(x)={\sin }^3\left(\frac{\pi }{3}\cos \left(\frac{\pi }{3\sqrt{2}}(-4x^3+5x^2+1{)}^{3/2}\right)\right)$. Then, at $x=1$,

The Art of Decomposition: Taming the Monster. Welcome, fellow traveler on the path to JEE mastery. Today, we face a function that, at first glance, seems designed to induce panic.

It is a nested, multi-layered beast:

But here is the secret: complexity is often just a mask for simplicity. The first step in our journey is to strip away that mask. We define the innermost algebraic core as:

By doing this, we transform our terrifying function into a much friendlier form:

We have effectively turned a mountain into a series of small, manageable hills.

Evaluating at $x=1$

Now, we need to see what happens at the specific point $x=1$. We start by evaluating our inner function $g(1)$. Substituting $x=1$ into our expression, we get:

The polynomial inside the bracket simplifies beautifully: $-4+5+1=2$. So, we have:

Since $2^{3/2}=2\sqrt{2}$, the expression becomes:

With this, we can find $y(1)={\sin }^3\left(\frac{\pi }{3}\cos (2\pi /3)\right)$. Since $\cos (2\pi /3)=-1/2$, we have:

We have successfully conquered the first peak.

Now, we must differentiate. This is where the Chain Rule becomes our best friend. We need $y^{\prime }(x)$.

Differentiating $y={\sin }^3(\dots )$ requires us to peel the layers: first the cube, then the sine, then the cosine, and finally the inner function $g(x)$. The derivative is:

Before we plug in $x=1$, we need $g^{\prime }(1)$. Differentiating $g(x)=\frac{\pi }{3\sqrt{2}}(-4x^3+5x^2+1{)}^{3/2}$ gives:

At $x=1$, this simplifies to:

Finally, we bring it all together. Substituting $g(1)=2\pi /3$ and $g^{\prime }(1)=-\pi$ into our derivative expression, we calculate:

Using the standard values $\sin (-\pi /6)=-1/2$, $\cos (-\pi /6)=\sqrt{3}/2$, and $\sin (2\pi /3)=\sqrt{3}/2$, we get:

Now, we check the relation:

The beauty of the cancellation is the reward for your persistence. You have solved the problem not by brute force, but by elegant, systematic decomposition. The final result is 0.