Animated Solution for Mathematics - Differentiation: Let $f(x)=x^5+2e^{x/4}$ for all $x\in \mathbb{R}$. Consider a function $g(x)$ such that $(g\circ f)(x)=x$ for all $x\in \mathbb{R}$. Then the value of $8g^{\prime }(2)$ is :

Welcome, future engineer! Today, we are going to dismantle a problem that, at first glance, looks like a monster. We are given $f(x)=x^5+2e^{\frac{x}{4}}$ and asked to find the value of $8g^{\prime }(2)$, where $g(x)$ is the inverse of $f(x)$.

Many students see the $x^5$ and the exponential term $2e^{\frac{x}{4}}$ and immediately freeze, thinking, "How on earth do I invert this?"

Here is the secret: You don't. In the world of JEE Advanced, you are rarely asked to perform brute-force algebra when a deeper, more elegant property exists. We are going to use the power of the Chain Rule and the geometric beauty of inverse functions to solve this without ever finding the explicit form of $g(x)$.

The Fundamental Identity

Let us start with the definition of an inverse function. If $g(x)$ is the inverse of $f(x)$, then by definition, $g(f(x))=x$. This is the bedrock of our solution.

Imagine you are standing on the graph of $f(x)$. If you input $x$, you get $f(x)$. If you then feed that result into $g(x)$, you are brought right back to your starting point, $x$.

Now, let us differentiate both sides of this identity with respect to $x$. Using the Chain Rule on the left side, we get:

This is our golden formula. It tells us that the derivative of the inverse function at the point $f(x)$ is simply the reciprocal of the derivative of the original function at $x$.

Mathematically, we can rearrange this to isolate the term we need:

This is profound. It means that to find the slope of the inverse function at a specific point, we only need to know the slope of the original function at the corresponding point. We do not need to know what $g(x)$ looks like; we only need to know where it came from.

The Inspection Trap

We need to find $g^{\prime }(2)$. Looking at our formula, this implies that $f(x)$ must equal $2$ for some value of $x$. So, we set up the equation:

I know what you are thinking: "This is a transcendental equation! How do I solve for $x$?" This is where the JEE examiner tests your intuition.

When you see a mix of polynomials and exponentials, do not reach for your calculator or complex numerical methods. Reach for your intuition. Test the simple integers. Let us try $x=0$:

It works! By inspection, we have found that $f(0)=2$. This means that the point $(0,2)$ lies on the graph of $f(x)$.

Consequently, the point $(2,0)$ must lie on the graph of $g(x)$. We have successfully mapped our target point without any complex algebra.

The Final Calculation

Now that we know $f(0)=2$, our derivative formula becomes:

We need to find $f^{\prime }(x)$. Let us differentiate $f(x)=x^5+2e^{\frac{x}{4}}$ term by term. Using the power rule for the first term and the chain rule for the exponential term, we get:

Simplifying this, we have:

Now, let us evaluate this at $x=0$:

We are almost at the finish line. Plugging this back into our inverse derivative formula:

The question asks for $8g^{\prime }(2)$. So, we simply calculate:

Look at what we have achieved. We navigated a seemingly impossible equation by relying on the fundamental properties of inverse functions and the elegance of the Chain Rule.

We didn't need to solve for $g(x)$; we only needed to understand its relationship to $f(x)$. This is the essence of JEE Advanced mathematics: it is not about brute force; it is about finding the most elegant path to the truth.

Keep practicing this mindset, and you will find that even the most intimidating problems become beautiful puzzles waiting to be solved. The final answer is 16.