Animated Solution for Mathematics - Differentiation: Let $f$ and $g$ be differentiable functions on $R$, such that $fog$ is the identity function. If for some $a,b\in R,g^{\prime }(a)=5$ and $g(a)=b$, then $f^{\prime }(b)$ is equal to :

Understanding Inverse Derivatives

Welcome, future engineer! Today, we are going to peel back the curtain on one of the most elegant relationships in calculus: the derivative of inverse functions.

Often, students see a problem involving $f(g(x))=x$ and panic, trying to find the explicit forms of $f$ and $g$. But here is the secret: you don't need them. You only need the geometry of the relationship.

The Round Trip

Imagine you are standing at a point $a$ on the real number line. You apply a function $g$, which transports you to a new location, $b$.

Now, you apply another function $f$ to your new location $b$. The problem tells us that $f\circ g$ is the identity function. This means that after this entire journey—applying $g$ and then $f$—you land exactly back where you started: at $a$.

Mathematically, we write this as:

This is our anchor. It tells us that $f$ and $g$ are essentially undoing each other's work. They are inverses.

The Power of the Chain Rule

Now, we want to find $f^{\prime }(b)$. We know the derivative of the inner function $g$ at $a$, which is $g^{\prime }(a)=5$.

How do we bridge the gap between $f^{\prime }(b)$ and $g^{\prime }(a)$? We use the Chain Rule. If we differentiate both sides of our identity equation with respect to $x$, we get:

Applying the chain rule to the left side, we obtain:

This equation is the heart of the problem. It tells us that the rate of change of the composite function is always unity, because the output $x$ changes at the same rate as the input $x$.

The Substitution Dance

We are given that $g(a)=b$. Let's substitute $x=a$ into our derivative equation:

Since $g(a)=b$, we can replace $g(a)$ with $b$:

We are almost there! We know $g^{\prime }(a)=5$. Plugging this in, we get:

Finally, we isolate $f^{\prime }(b)$ by dividing both sides by $5$:

Look at that! We didn't need to know what $f$ or $g$ actually were. We only needed to understand how they interact.

This reciprocal relationship, $f^{\prime }(b)=\frac{1}{g^{\prime }(a)}$, is a fundamental concept in calculus. Whenever you see inverse functions, remember this symmetry. It is not just math; it is the geometry of undoing.