Animated Solution for Mathematics - Differentiation: If $f(1)=1,f^{\prime }(1)=3$, then the derivative of $f(f(f(x)))+(f(x){)}^2$ at $x=1$ is :

Peeling the Mathematical Onion

Welcome, fellow traveler on the road to JEE Advanced excellence! Today, we are going to demystify a problem that often intimidates students: the differentiation of nested composite functions.

When you see an expression like $g(x)=f(f(f(x)))+(f(x){)}^2$, it is easy to feel overwhelmed. But I want you to take a deep breath. In calculus, complexity is often just a series of simple steps hidden behind a curtain of notation. Let us pull back that curtain.

Peeling the Onion

Think of the Chain Rule as the art of peeling an onion. You have layers, and you must respect each one. Our function $g(x)$ has two main terms.

Let us focus on the first, the triple-nested $f(f(f(x)))$. To find its derivative, we start from the outside. The derivative of the outermost $f$ is $f^{\prime }$, and we keep the entire 'inside' exactly as it is. This gives us $f^{\prime }(f(f(x)))$.

But we are not done! We must now multiply by the derivative of the next layer, which is the middle $f$. This gives us $\cdot f^{\prime }(f(x))$.

Finally, we reach the core, the innermost $f(x)$, and multiply by its derivative, $f^{\prime }(x)$. Putting it all together, the derivative of the first term is $f^{\prime }(f(f(x)))\cdot f^{\prime }(f(x))\cdot f^{\prime }(x)$. It looks like a long chain, but it is perfectly logical. Each multiplication represents the rate of change of that specific layer.

The Power Rule and the Inner Function

Now, let us turn our attention to the second term: $(f(x){)}^2$. This is a classic application of the Power Rule.

The exponent $2$ comes down to the front, and the power decreases by one, leaving us with $2(f(x){)}^1$. However, we must not forget the Chain Rule here as well!

The 'inside' is $f(x)$, so we must multiply by the derivative of the inside, $f^{\prime }(x)$. Thus, the derivative of the second term is $2f(x)\cdot f^{\prime }(x)$.

The Moment of Truth

We have successfully constructed the derivative of our entire function $g(x)$:

Now, we arrive at the most satisfying part of the problem: the evaluation at $x=1$. We are given $f(1)=1$ and $f^{\prime }(1)=3$. Let us substitute $x=1$ into our expression:

Because $f(1)=1$, the nested terms collapse beautifully. $f(1)$ is $1$, so $f(f(1))$ is $f(1)$, which is $1$. Consequently, $f(f(f(1)))$ is also $1$.

Our expression simplifies to:

Since $f^{\prime }(1)=3$, we have:

And there you have it! The final answer is 33.

What seemed like a terrifying wall of functions was actually just a sequence of careful, methodical steps. Remember, in JEE Advanced, the goal is not to memorize formulas, but to understand the flow of the math.

When you encounter nested functions, do not panic—just start peeling the onion, one layer at a time. You have the tools; now go forth and conquer!