Problem 3 - Calculus Early Transcendentals

Question

Let \(f(x)=x^{5}\) and \(g(x)=2 x-3\) (a) Find \((f \circ g)(x)\) and \((f \circ g)^{\prime}(x)\) (b) Find \((g \circ f)(x)\) and \((g \circ f)^{\prime}(x)\)

Step-by-Step Solution

Answer

(a) \((f \circ g)(x) = (2x-3)^5\), \((f \circ g)'(x) = 10(2x-3)^4\); (b) \((g \circ f)(x) = 2x^5-3\), \((g \circ f)'(x) = 10x^4\).

Step 1: Understand the Composition Notation

The notation \((f \circ g)(x)\) signifies the composition of the functions \(f\) and \(g\), which means \(f(g(x))\). Conversely, \((g \circ f)(x)\) means \(g(f(x))\).

Step 2: Compute \((f \circ g)(x)\)

Substitute the expression for \(g(x)\) into \(f(x)\). Since \(g(x) = 2x - 3\) and \(f(x) = x^5\), you have:\[(f \circ g)(x) = f(g(x)) = f(2x - 3) = (2x - 3)^5\]

Step 3: Differentiate \((f \circ g)(x)\)

To find \((f \circ g)'(x)\), differentiate \((2x - 3)^5\) using the chain rule. Let \(u = 2x - 3\), then \(f(u) = u^5\), and\[\frac{d}{dx}((2x - 3)^5) = 5(2x - 3)^4 \times 2 = 10(2x - 3)^4\]

Step 4: Compute \((g \circ f)(x)\)

Substitute the expression for \(f(x)\) into \(g(x)\). Using \(f(x) = x^5\) and \(g(x) = 2x - 3\), you get:\[(g \circ f)(x) = g(f(x)) = g(x^5) = 2(x^5) - 3 = 2x^5 - 3\]

Step 5: Differentiate \((g \circ f)(x)\)

Find the derivative of \(2x^5 - 3\) using basic differentiation rules:\[\frac{d}{dx}(2x^5 - 3) = 10x^4\]

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus, used to determine the rate at which a function is changing at any given point. When you're finding the derivative of a function, you're identifying the slope of the tangent line to the function's graph at a specific point. This can provide insights into the function's behavior—such as where it is increasing or decreasing.

• The most common notation for a derivative is \( f'(x) \), which denotes the derivative of \( f(x) \).
• Taking a derivative involves applying rules like the power rule, where \( \frac{d}{dx}(x^n) = nx^{n-1} \).
• In our exercise, we find derivatives of composed functions, which involves techniques like the chain rule.

Chain Rule

The chain rule is a crucial differentiation technique used for finding the derivative of composite functions. A composite function is one where you have a function inside another function, which means you compose two functions.
For example, in our exercise where we find the derivative of \((f \circ g)(x) = (2x - 3)^5 \), we're tasked with finding \((f \circ g)'(x)\).

• First, identify the inner and outer functions. Here, \( u = 2x - 3 \) is the inner function and \( u^5 \) is the outer.
• Apply the chain rule: differentiate the outer function with respect to the inner function \( u \), then multiply by the derivative of the inner function \( u \).
• So, the derivative becomes \( 5(2x - 3)^4 \times 2 = 10(2x - 3)^4 \), illustrating how the chain rule allows us to tackle more complex differentiation tasks.

Function Notation

Function notation allows us to clearly express the dependencies between variables and it is particularly useful when dealing with composite functions.
It is denoted as \( f(x) \), indicating each input \( x \) maps to a unique output through the function \( f \). Function composition, denoted by \((f \circ g)(x)\), is a way of "nesting" these functions so that the output of one function becomes the input of another.

• For \((f \circ g)(x)\), you compute this by substituting \( g(x) \) into \( f(x) \), making it \( f(g(x)) \).
• Conversely, \((g \circ f)(x)\) involves substituting \( f(x) \) into \( g(x) \), resulting in \( g(f(x)) \).
• This notation helps in organizing functions and is essential when performing operations such as differentiation of composed functions.