Animated Solution for Mathematics - Differentiation: Let $f:R\to R$ be defined as $f(x)=x^3+x-5$. If $g(x)$ is a function such that $f(g(x))=x,\forall x\in R$, then $g^{\prime }(63)$ is equal to

Welcome, fellow traveler on the path to JEE mastery. Today, we are going to unravel a problem that, at first glance, might seem like a daunting algebraic puzzle.

We are given a cubic function, $f(x)=x^3+x-5$, and we are asked to find the derivative of its inverse, $g(x)$, at a specific point. This is a classic scenario where brute force will lead you into a labyrinth, but conceptual elegance will guide you straight to the exit.

Understanding the Inverse

We are given the identity $f(g(x))=x$. In the language of functions, this is the definition of an inverse.

Imagine $f$ as a machine that takes an input and transforms it. If $g$ is the inverse, it is the machine that perfectly reverses that transformation.

When you feed $x$ into $g$, you get some value $a$, and when you feed $a$ into $f$, you get $x$ back. This is the fundamental symmetry of inverse functions.

The Derivative Bridge

How do we find the derivative of this 'undoing' machine? We use the chain rule.

If we differentiate both sides of the identity $f(g(x))=x$ with respect to $x$, we get:

Applying the chain rule to the left side, we obtain $f^{\prime }(g(x))\cdot g^{\prime }(x)=1$. Rearranging this gives us the powerful, standard formula for the derivative of an inverse function:

This formula is a cornerstone of calculus. It tells us that the slope of the inverse function at a point is the reciprocal of the slope of the original function at the corresponding point.

The Search for the Hidden Value

We need to find $g^{\prime }(63)$. Using our formula, this becomes:

To solve this, we need the value of $g(63)$. Let's call this unknown value $a$.

By the definition of an inverse, $g(63)=a$ implies $f(a)=63$. Substituting this into our original function, we get:

Now, I know what you are thinking: "Do I need to use Cardano's formula for a cubic?" Absolutely not!

In the context of JEE, these equations are designed to have a clean, integer solution. Let's test small integers.

If $a=4$, then $4^3+4-68=64+4-68=0$. It works! So, $g(63)=4$.

The Final Calculation

Now that we have $g(63)=4$, we just need to find $f^{\prime }(4)$. Our derivative function is $f^{\prime }(x)=3x^2+1$.

Substituting $x=4$:

Finally, we plug this back into our inverse derivative formula:

And there you have it. The slope of the inverse function at $x=63$ is $\frac{1}{49}$.

We didn't need to solve the cubic for $x$, nor did we need to find the explicit form of $g(x)$. We simply used the symmetry of the functions and the power of the chain rule. Keep this logic in your toolkit—it is a weapon that will serve you well in the exam hall.