Animated Solution for Mathematics - Differentiation: If $g$ is the inverse of a function $f$ and $f^{\prime }(x)=\frac{1}{1+x^5}$, then $g^{\prime }(x)$ is equal to

Imagine you are standing in front of a mirror. You raise your right hand, and your reflection raises its left. There is a perfect, elegant symmetry between you and your reflection.

In the world of mathematics, inverse functions share this exact kind of relationship. When we talk about a function $f(x)$ and its inverse $g(x)$, we are talking about two entities that are mirror images of each other across the line $y=x$.

This isn't just a pretty geometric fact; it is the key to unlocking the derivative of the inverse function without ever needing to solve for the inverse itself.

To begin our journey, we must anchor ourselves in the definition of an inverse function. If $g$ is the inverse of $f$, then applying $f$ to $g(x)$ should return us exactly to where we started: $x$.

Mathematically, we write this as:

This is our bedrock and the most important equation in this problem. It tells us that the composition of a function and its inverse is the identity function. If you ever feel lost, come back to this identity; it is the bridge between the two functions.

Now, we want to find $g^{\prime }(x)$, the derivative of the inverse function. We have our identity $f(g(x))=x$. Let's differentiate both sides with respect to $x$.

On the right side, the derivative of $x$ is simply $1$. On the left side, we encounter a composite function. We apply the Chain Rule: the derivative of $f(g(x))$ is $f^{\prime }(g(x))\cdot g^{\prime }(x)$.

So, our equation becomes:

This is the moment of truth. We have successfully linked the derivative of the inverse function to the derivative of the original function.

We are given that:

Now, look closely at our derived formula: $g^{\prime }(x)=\frac{1}{f^{\prime }(g(x))}$. We need to evaluate $f^{\prime }$ at $g(x)$.

This means we take our expression for $f^{\prime }(x)$ and replace every instance of $x$ with $g(x)$:

It is a simple substitution, but it is the heart of the problem. We are not changing the function; we are just changing the input to match the needs of our derivative formula.

Finally, we substitute this back into our expression for $g^{\prime }(x)$:

When we have a fraction in the denominator, it flips to the numerator. The expression simplifies beautifully to:

And there it is! We have found the derivative of the inverse function without ever needing to find the inverse function itself.

This is the elegance of calculus. We didn't need to struggle with algebraic inversions; we only needed to understand the relationship between the functions and their derivatives. Remember this technique—it is a favorite of JEE examiners because it tests your conceptual depth rather than your ability to perform tedious algebra. You have mastered the mirror world of functions!