Problem 3 - Calculus Early Transcendentals

Question

Fill in the blanks. The derivative of \(f(g(x))\) equals \(f^{\prime}\) evaluated at ________ multiplied by \(g^{\prime}\) evaluated at ________

Step-by-Step Solution

Answer

Question: Determine the expression for the derivative of \(f(g(x))\), and specify where to evaluate each derivative. Answer: The derivative of \(f(g(x))\) equals \(f^{\prime}\) evaluated at \(g(x)\) multiplied by \(g^{\prime}\) evaluated at \(x\).

Step 1: Identify the inner and outer functions

First, we need to identify the inner and outer functions in the composite function \(f(g(x))\). Here, the inner function is \(g(x)\) and the outer function is \(f(u)\), where \(u = g(x)\).

Step 2: Apply the chain rule

To differentiate \(f(g(x))\) with respect to \(x\), we apply the chain rule, which states that \((f(g(x)))'=f'(g(x))\cdot g'(x)\). This means we need to find the derivatives of \(f(u)\) and \(g(x)\), then multiply them together.

Step 3: Determine where to evaluate the derivatives

Now, we need to specify where to evaluate each derivative. According to the chain rule, we need to evaluate \(f'(u)\) at \(u = g(x)\), which is the output of the inner function, and evaluate \(g'(x)\) at the input \(x\). So the blank spaces should be filled as follows: The derivative of \(f(g(x))\) equals \(f^{\prime}\) evaluated at \(g(x)\) multiplied by \(g^{\prime}\) evaluated at \(x\).

Key Concepts

Composite Function

A composite function is when one function is applied to the result of another function. With composite functions, you are essentially nesting one function inside another. For instance, if we have two functions, say \( f(x) \) and \( g(x) \), the composite function \( f(g(x)) \) means that you first apply \( g(x) \), and then use this result as the input for \( f(x) \).

Using composite functions allows for the creation of new functions that can model more complex behaviors by combining simpler, existing functions. In the composite function \( f(g(x)) \), \( g(x) \) is known as the "inner function," while \( f(u) \), where \( u = g(x) \), is the "outer function."

• Inner Function: Performs the first operation
• Outer Function: Takes the result of the inner function as its input

Recognizing these parts is crucial when applying other operations, like finding derivatives.

Derivative

The derivative of a function measures how that function's output value changes as its input value changes. It gives us the "slope" of the function at any point, showing how fast the function is climbing or descending.

For a given function \( y = f(x) \), the derivative \( f'(x) \) tells us the rate at which \( y \) changes with respect to \( x \). Calculating derivatives is especially important in understanding how functions behave and in solving real-world problems involving rates of change.

When dealing with composite functions, such as \( f(g(x)) \), calculating derivatives requires using the chain rule, which takes into account both the rate of change of \( f \) with respect to \( g \) and \( g \) with respect to \( x \).

Remember:

• Derivatives are like the slope at any given point
• They help in predicting how changes in \( x \) affect \( y \)
• Important: Derivatives are central to calculus and solving dynamic problems

Differentiation

Differentiation is the process of finding a derivative. It involves mathematical techniques to determine the rate at which a function is changing at any given point. It's a fundamental tool in calculus, used to understand variable relationships and model complex systems.

When you differentiate a composite function \( f(g(x)) \), you use the **chain rule**. The chain rule is a formula for computing the derivative of the composition of two or more functions. Think of it as taking a small change in \( x \) and seeing how that change propagates through \( g(x) \), then through \( f(u) \). The rule can be described as:

\[(f(g(x)))' = f'(g(x)) \cdot g'(x) \]
This means you first take the derivative of the outer function \( f \), evaluated at the inner function \( g(x) \), and multiply it by the derivative of the inner function \( g(x) \) itself.

• Differentiation helps in breaking down complex expressions
• It's used extensively in physics, engineering, economics, and beyond
• With the chain rule, you look at how changes progress through multiple functions