Problem 81 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(f(0)=0\) and \(f^{\prime}(0)=2\). Find the derivative of \(f(f(f(f(x))))\) at \(x=0 .\)

Step-by-Step Solution

Answer

The derivative at \( x = 0 \) is 16.

Step 1: Understand the Problem

We are tasked with finding the derivative of the nested function \( f(f(f(f(x)))) \) at \( x = 0 \). Given are \( f(0) = 0 \) and \( f'(0) = 2 \).

Step 2: Use the Chain Rule Multiple Times

We will use the chain rule to find the derivative. The chain rule states that the derivative of \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). Since the function is nested four times, we need to apply the chain rule repeatedly.

Step 3: Apply the Chain Rule Between First and Second Layer

Consider \( y = f(f(f(x))) \). The derivative \( \frac{dy}{dx} \) is \( f'(f(f(x))) \cdot f'(f(x)) \cdot f'(x) \cdot f'(x) \), considering the application of chain rule through each nested layer.

Step 4: Evaluate at \( x = 0 \)

Substitute \( x = 0 \) into the expression derived from the chain rule: \( \frac{d}{dx}[f(f(f(f(x))))]_{x=0} = f'(f(f(f(0)))) \cdot f'(f(f(0))) \cdot f'(f(0)) \cdot f'(0) \). Since \( f(0) = 0 \), all nested \( f(x) \) evaluations result in 0. So, we get \( f'(0) \cdot f'(0) \cdot f'(0) \cdot f'(0) \).

Step 5: Compute the Final Derivative

Insert the value that was given: \( f'(0) = 2 \). Therefore, the expression simplifies to \( 2 \cdot 2 \cdot 2 \cdot 2 = 16 \).

Step 6: Conclusion

The derivative of \( f(f(f(f(x)))) \) at \( x = 0 \) is \( 16 \).

Key Concepts

Understanding Derivatives

Derivatives are fundamental in calculus and represent how a function changes as its input changes. In simple terms, the derivative tells us the rate of change or slope of the function at any given point.
In the context of the exercise, we want to understand the change in the nested function \(f(f(f(f(x))))\) at \(x = 0\). At this point, we're primarily interested in finding out how this deeply nested function behaves as we input and begin close to the value of zero.

• Use derivatives to predict future behavior of near points.
• Essential for various applications in physics, engineering, and economics where rates of change are explored.

When solving a problem like this, where multiple functions are nested within each other, understanding the individual derivative notation, such as \(f'(x)\), helps to track which function is being differentiated. This is crucial when we move onto the concept of nested functions.

Decoding Nested Functions

Nested functions occur when a function is placed inside another function. An example is \(f(g(x))\), where \(g(x)\) is first evaluated, and its result is then used as the input for \(f\). In our problem, we have several layers, specifically four, nested as \(f(f(f(f(x))))\). This requires careful consideration of each layer.

• Break down each function: Evaluate layer by layer.
• Track your progress and ensure each function's output serves as the input to the next.

Nest functions deeply challenge our ability to keep track of what goes where. Yet, through practice and careful application of calculus rules, students can decipher each complexity. This is where we apply the chain rule to help simplify the unraveling of these functions.

Calculus Problem Solving with the Chain Rule

The chain rule in calculus is a powerful tool for finding derivatives of composite functions, which are functions made by combining other functions. The rule tells us how to differentiate a function that is composed of other functions, like our nested scenario \(f(f(f(f(x))))\).
The chain rule is described as: If you have a function \(y = f(g(x))\), then the derivative \( \frac{dy}{dx} \) is \( f'(g(x)) \cdot g'(x) \). This helps us break complex differentiation tasks into simpler steps.
In our exercise:

• Start from the innermost function and move outward.
• Apply the chain rule layer by layer.
• Utilize given values like \(f(0) = 0\) and \(f'(0) = 2\) to shift from theory to tangible computation.

Upon completion, applying these rules step by step leads to the final result. In this case, repeated application of the chain rule resulted in a derivative of \(16\) at \(x = 0\). Knowing when to employ the chain rule effectively streamlines the process and helps solve complex calculus problems with ease.