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

Step 1: Understand the problem

We are given that the function \(f\) satisfies \(f(0) = 0\) and \(f'(0) = 2\). We need to find the derivative of the composition of \(f\) applied four times, \(f(f(f(f(x))))\), at \(x = 0\).

Step 2: Chain Rule for Composition

To find the derivative of the composition \(f(f(f(f(x))))\), we need to apply the chain rule. According to the chain rule, \((g(h(x)))' = g'(h(x))h'(x)\). For four functions, it would be similar: \[ \frac{d}{dx} f(f(f(f(x)))) = f'(f(f(f(x)))) \cdot f'(f(f(x))) \cdot f'(f(x)) \cdot f'(x). \]

Step 3: Apply Initial Conditions

By the given conditions, \(f(0) = 0\) and \(f'(0) = 2\). Since \(f(0) = 0\), substituting \(0\) into the derivatives when evaluating at \(x = 0\) simplifies further terms to: \[ f'(f(f(f(0)))) = f'(f(f(0))) = f'(f(0)) = f'(0). \]

Step 4: Calculate Derivative Specifically at x = 0

Using the simplifications, the expression becomes: \[ f'(f(f(f(0)))) \cdot f'(f(f(0))) \cdot f'(f(0)) \cdot f'(0) = (f'(0))^4. \] We know from the initial conditions \(f'(0) = 2\), so: \[ (f'(0))^4 = 2^4 = 16. \]

Step 5: Conclusion

Hence, the derivative of \(f(f(f(f(x))))\) at \(x = 0\) is 16.

Key Concepts

Derivative

In calculus, the derivative of a function at a particular point tells you the rate at which the function's output value changes with respect to its input. It's like asking, "How quickly is the function climbing or falling at this particular spot?" For a function \( f(x) \), the derivative is often denoted as \( f'(x) \) or \( \frac{df}{dx} \). In simpler terms, the derivative measures the slope of the tangent line to the curve of the function at any given point.

To calculate a derivative, you might use techniques like power rules or product rules, but in our scenario, we're dealing with function compositions and need the chain rule. Important tips for derivatives include:

• Think of it as capturing the essence of change in a function.
• If a function's slope is positive, it's increasing.
• If the slope is negative, it's decreasing.
• A zero slope indicates no change at that instant.

Understanding derivatives is crucial because they play a key role in analyzing the behavior of functions.

Chain Rule

The Chain Rule is an essential tool in calculus for finding the derivative of composite functions. Composite functions are functions within functions, like our example \( f(f(f(f(x)))) \). The chain rule allows us to systematically "break it down" into simpler pieces we can handle. Chains Rule states: if you have a composition of functions \( g(h(x)) \), the derivative is given by \( g'(h(x)) \cdot h'(x) \).

When you apply the chain rule:

• Start by identifying the outer and inner functions.
• Take the derivative of the outer function, keeping the inside unchanged.
• Multiply by the derivative of the inner function.

This method helps us find the derivative of \( f(f(f(f(x)))) \) efficiently by breaking it into manageable pieces at \( x = 0 \). Using the chain rule repeatedly gives insight into how deeply nested functions relate to each other and how their changes cascade.

Function Composition

Function Composition in calculus involves creating a new function by combining one or more functions. Think of it as plugging one function into another. For instance, in our task, we are dealing with \( f(f(f(f(x)))) \). This means the output of one \( f(x) \) becomes the input of the next.

To better understand function composition:

• Consider it like gears in machines; if one turns, it makes the next one turn.
• Writing \( f(g(x)) \) is akin to \( f \) operating on the result of \( g \).
• Helps in breaking complex functions into smaller parts.

This concept is powerful because it neatly encapsulates processes in mathematics, making them much easier to analyze, especially when applying the chain rule.

Initial Conditions

Initial Conditions are the values we know beforehand that help solve a problem or equation. In calculus, these values are crucial for determining specific solutions among many possibilities. For instance, in our equation, knowing \( f(0) = 0 \) and \( f'(0) = 2 \) provides essential grounding points.

Why are initial conditions important? Here’s a quick overview:

• They serve as anchor points to start calculating derivatives or integrals.
• Allows specificity in solutions, pinning down exact numbers instead of general forms.
• Guides understanding of how a function behaves at those specific points.

Initial conditions are akin to a road map, giving direction and accuracy, ensuring that calculations lead to precise and meaningful results, as seen in evaluating \( (f'(0))^4 \) to get the derivative at \( x = 0 \).