Animated Solution for Mathematics - Differentiation: Let $f:(-1,1)\to R$ be a differentiable function with $f(0)=-1$ and $f^{\prime }(0)=1$. Let $g(x)=[f(2f(x)+2){]}^2$. Then $g^{\prime }(0)=$

Imagine you are standing before a complex machine. It has gears within gears, and levers that trigger other levers. In mathematics, we call this a composite function.

The problem at hand, $g(x)=[f(2f(x)+2){]}^2$, is exactly that—a mathematical machine. When you first look at it, it is easy to feel intimidated.

But remember, every complex machine is just a collection of simple parts working in harmony. Our goal is to find the rate of change, $g^{\prime }(0)$, which tells us how the output of this machine shifts when we nudge the input $x$ at the zero mark.

To solve this, we use the Chain Rule, which I like to think of as a relay race. Each function in the composition is a runner, and the derivative is the baton being passed from one to the next.

We start with the outermost runner: the square function. The derivative of $u^2$ is $2u$. So, our first step is to write:

We have successfully passed the baton!

Now, we look at the next runner: the function $f$ with the inner argument $v=2f(x)+2$. The derivative of $f(v)$ is $f^{\prime }(v)\cdot v^{\prime }$.

This gives us the next piece of our puzzle: $f^{\prime }(2f(x)+2)\cdot \frac{d}{dx}(2f(x)+2)$. Finally, we reach the innermost runner, the linear expression $2f(x)+2$. Its derivative is simply $2f^{\prime }(x)$.

Now, we bring all our runners together. Multiplying these pieces, we get the complete expression for the derivative:

Simplifying the constants, we get:

This is the master formula for our machine. It looks intimidating, but notice how structured it is. It is just a product of the function's value, its derivative at the transformed point, and the derivative at the original point.

Now, we evaluate this at $x=0$. We are given $f(0)=-1$ and $f^{\prime }(0)=1$.

Let's look at that inner argument: $2f(0)+2$. Substituting $f(0)=-1$, we get $2(-1)+2=0$.

This is the moment of magic! The entire complex nested argument collapses into zero. Our expression becomes:

Substituting the known values, we have $4\cdot (-1)\cdot 1\cdot 1$, which equals $-4$.

The machine has been decoded, and the final answer is $-4$. You see, the complexity was just a veil; by breaking it down step by step, we found the elegant simplicity hidden underneath.