Problem 53 - Calculus Single Variable

Question

Let \(c=4, F(x)=1+3 x, G(x)=\sqrt{x}\) and \(H(x)=x /(x+5) .\) Calculate the requested derivative. $$ (G \circ F \circ F)^{\prime}(c) $$

Step-by-Step Solution

Answer

The derivative \((G \circ F \circ F)'(4) = \frac{9\sqrt{10}}{40}.\)

Step 1: Understand the Function Composition

We need to compute the derivative of the composed function \( (G \circ F \circ F)(x) \). This means \( G(F(F(x))) \). Let's break it down: First, apply \( F(x) \) to \( c \), then apply \( F \) again to the result, and finally, apply \( G \).

Step 2: Evaluate the Inner Functions

First, find \( F(F(x)) \). As \( F(x) = 1 + 3x \), we evaluate \( F(x) \) twice: first \( F(x) = 1 + 3x \), then apply \( F \) again: \( F(F(x)) = F(1 + 3x) = 1 + 3(1 + 3x) = 1 + 3 + 9x = 4 + 9x \).

Step 3: Apply the Outer Function

Now evaluate \( G(F(F(x))) \). Since \( G(x) = \sqrt{x} \), then \( G(F(F(x))) = \sqrt{4 + 9x} \).

Step 4: Differentiate the Composed Function

Now differentiate \( G(F(F(x))) = \sqrt{4 + 9x} \). The derivative of \( \sqrt{u} \) is \( \frac{1}{2\sqrt{u}} \cdot u' \). Thus, \( \frac{d}{dx} G(F(F(x))) = \frac{1}{2}\sqrt{4 + 9x} \cdot (9) = \frac{9}{2\sqrt{4 + 9x}} \).

Step 5: Evaluate the Derivative at \( x = c \)

Substitute \( x = 4 \) into the derivative: \( \frac{9}{2\sqrt{4 + 9(4)}} = \frac{9}{2\sqrt{4 + 36}} = \frac{9}{2\sqrt{40}} = \frac{9}{2 \times 2\sqrt{10}} = \frac{9}{4\sqrt{10}} \). Simplifying gives \( \frac{9}{4\sqrt{10}} = \frac{9\sqrt{10}}{40} \) after rationalizing the denominator.

Key Concepts

Function Composition

When we talk about function composition, we are combining two or more functions into a single function. It’s a bit like layers of an onion; you peel back one layer to get to the next. For instance, if you have two functions, such as \( F(x) \) and \( G(x) \), and you create a composition \( G(F(x)) \), you are essentially first applying the function \( F \) and then applying \( G \) to the result of \( F \). This process allows complex functions to be broken down into simpler parts.

In our exercise, we have \( (G \circ F \circ F)(x) \), indicating the composition of three functions. This means we apply \( F \) to \( x \), apply \( F \) again to the result, and finally apply \( G \). Here’s a quick breakdown of the compositional layers:

• Start with the innermost function \( F(x) = 1 + 3x \).
• Apply \( F \) again to \( F(x) \), forming \( F(F(x)) \).
• Finally, apply \( G \) to \( F(F(x)) \), resulting in \( G(F(F(x))) \).

By understanding this layered structure, you now hold a tool to approach more intricate functions with ease.

Derivative of Composite Functions

Finding the derivative of composite functions can seem daunting, but it becomes manageable when using the Chain Rule. The Chain Rule is a formula used to compute the derivative of a composition of two or more functions. If you have a composite function \( y = G(F(x)) \), then the derivative of \( y \) with respect to \( x \) can be found using this rule.

The Chain Rule is expressed simply as:\( (G \circ F)'(x) = G'(F(x)) \cdot F'(x) \). This means you first differentiate the outer function \( G \) evaluated at the inner function \( F(x) \), and then multiply by the derivative of the inner function \( F \).

In the provided solution, let’s break down the steps:

• First, determine the derivative of the outermost function, \( G(x) = \sqrt{x} \), which is \( G'(x) = \frac{1}{2\sqrt{x}} \).
• Next, evaluate this derivative at \( F(F(x)) \), which we found as \( 4 + 9x \).
• Then, find the derivative of this inner function, \( F(F(x)) = 4 + 9x \), which is \( 9 \).
• Finally, apply the Chain Rule: \( \frac{1}{2\sqrt{4 + 9x}} \cdot 9 \)

By following these guidelines, you’ll be able to navigate through derivative calculations of composite functions confidently.

Calculating Derivatives

Calculating derivatives is all about finding the rate of change. When dealing with complicated functions, such as composites, understanding their derivatives can reveal a lot about their behavior. Here’s how you approached the calculation in your exercise:

1. **Differentiate the Composite**: You employed the Chain Rule for \( G(F(F(x))) = \sqrt{4 + 9x} \). Remember, the key steps for achieving this are:

• Identify the derivative of the outer function \( G(x) = \sqrt{x} \), which gives \( \frac{1}{2\sqrt{x}} \).
• Apply it to the result of the inner function, producing \( \frac{1}{2\sqrt{4 + 9x}} \).
• Multiply by the derivative of the innermost function, which is \( 9 \).

2. **Evaluate at a Point**: Substitute \( x = 4 \) into the differentiated function to analyze the rate of change at this specific point:

• Insert \( x = 4 \) to find \( \frac{9}{2\sqrt{40}} \).
• Simplify this expression to \( \frac{9}{4\sqrt{10}} \) or \( \frac{9\sqrt{10}}{40} \) after rationalization.

Each derivative provides insight into how a function behaves as \( x \) changes. Understanding this process ensures you can tackle various derivative problems with confidence.