Animated Solution for Mathematics - Differentiation: Let $y(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. Then $y^{\prime }-y^{\prime \prime }$ at $x=-1$ is equal to

Imagine you are standing before a massive, intimidating product:

At first glance, it looks like a monster. If you try to attack this with the product rule, you will find yourself drowning in a sea of terms.

In the world of JEE Advanced, we don't fight monsters with brute force; we fight them with insight. Notice the exponents: $1,2,4,8,16$. They are powers of $2$, forming a classic geometric progression.

Whenever you see a product of terms like $(1+x)(1+x^2)(1+x^4)\dots$, your mind should immediately jump to the difference of squares identity: This is our secret weapon.

To unlock the potential of this identity, we perform a clever maneuver. We multiply and divide the entire expression by $(1-x)$.

Now, look at what happens: the numerator becomes $(1-x)(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. The first two terms collapse into $(1-x^2)$.

Then, $(1-x^2)(1+x^2)$ collapses into $(1-x^4)$. This chain reaction continues, with each step doubling the power of $x$, until the entire numerator simplifies to $1-x^{32}$.

Our function is now:

Now, we could use the quotient rule, but why make life difficult? Let's rearrange the equation: This gives us $y-xy=1-x^{32}$. This is much friendlier.

Differentiating with respect to $x$, we get:

Differentiating once more, we get:

We need to evaluate this at $x=-1$. First, $y(-1)=0$ because the original product contains $(1+x)$, which becomes $1-1=0$.

Substituting $x=-1$ and $y=0$ into our first derivative equation:

Now, for the second derivative:

This simplifies to $2y^{\prime \prime }(-1)=-960$, so $y^{\prime \prime }(-1)=-480$.

Finally, the question asks for $y^{\prime }-y^{\prime \prime }$ at $x=-1$. That is:

We have tamed the monster, and the result is a beautiful, clean integer. This is the power of pattern recognition in mathematics. The final answer is 496.