Animated Solution for Mathematics - Differentiation: If the function $f(x)=x^3+e^{x/2}$ and $g(x)=f^{-1}(x)$, then the value of $g^{\prime }(1)$ is

Welcome, future engineers! Today, we are tackling a problem that often intimidates students because it looks like a trap. We are given $f(x)=x^3+e^{x/2}$ and asked to find the derivative of its inverse, $g^{\prime }(1)$.

Many students immediately panic, trying to solve $y=x^3+e^{x/2}$ for $x$. Let me tell you right now: stop! You cannot isolate $x$ algebraically.

In the world of JEE Advanced, when a path seems impossible, it is usually because there is a more elegant, conceptual path waiting for you.

Your Secret Weapon

We don't need to know what $g(x)$ looks like to know how it changes. We have the Inverse Function Theorem, which is one of the most beautiful bridges between a function and its inverse.

It states that the derivative of the inverse function is the reciprocal of the derivative of the original function, evaluated at the corresponding point:

This is not just a formula; it is a geometric truth. If you visualize the graph of $f(x)$ and its reflection $g(x)$ across the line $y=x$, you will see that the tangent line to $f(x)$ at $(a,b)$ is simply the reflection of the tangent line to $g(x)$ at $(b,a)$.

The slope of one is the reciprocal of the other. It is elegant, it is simple, and it is powerful.

To find $g^{\prime }(1)$, we plug $x=1$ into our theorem:

We are stuck, aren't we? We don't know $g(1)$. But wait—let $g(1)=a$. By the definition of an inverse, this is equivalent to saying $f(a)=1$.

So, we need to solve:

In JEE problems, when you see a transcendental equation like this, do not reach for complex numerical methods. Look for the 'hidden' integer.

If we test $a=0$, we get $0^3+e^{0/2}=0+1=1$. It works!

Because $f(x)$ is strictly increasing (its derivative is always positive), we know $a=0$ is the only solution. Thus, $g(1)=0$.

Now, the path is clear. We need $f^{\prime }(0)$. Let's differentiate $f(x)=x^3+e^{x/2}$:

Applying the power rule and the chain rule, we get:

Evaluating this at $x=0$ is a breeze:

Finally, we return to our Inverse Function Theorem. Since $g^{\prime }(1)=\frac{1}{f^{\prime }(g(1))}$ and we know $g(1)=0$ and $f^{\prime }(0)=\frac{1}{2}$, we have:

There you have it! The answer is 2. We didn't need to invert the function; we just needed to understand the relationship between the slopes of the original and the inverse.

Keep this mindset for your JEE preparation: look for the symmetry, trust the theorems, and always check for simple values before diving into complex algebra. You have the tools—now go out there and conquer!