Problem 41 - Calculus Single Variable

Question

Use the specified value of \(c\) and the given information about \(f\) and \(g\) to compute \((g \circ f)^{\prime}(c)\). \(g(3)=2, g^{\prime}(2)=3, g^{\prime}(3)=4, f(2)=3, f^{\prime}(2)=8, f^{\prime}(3)=5,\) \(c=2\)

Step-by-Step Solution

Answer

The value of \((g \circ f)'(2)\) is 32.

Step 1: Understand the Composition Notation

The notation \((g \circ f)(x)\) means \(g(f(x))\). We are focused on finding \((g \circ f)'(c)\), which is the derivative of the composition function \(g(f(x))\) evaluated at \(x = c\).

Step 2: Apply the Chain Rule

To find the derivative of a composition of functions, use the chain rule, which states: \((g \circ f)'(x) = g'(f(x)) \cdot f'(x)\). Here, \(x = c = 2\).

Step 3: Substitute Known Values

First, evaluate \(f(c) = f(2)\), which is given as \(3\). Next, find \(g'(f(c)) = g'(3)\), which is given as \(4\). Finally, \(f'(c) = f'(2)\), which is given as \(8\).

Step 4: Compute the Derivative of the Composition

Substitute these values into the chain rule expression to get: \((g \circ f)'(2) = g'(f(2)) \cdot f'(2) = g'(3) \cdot 8\). Since \(g'(3) = 4\), compute \((g \circ f)'(2) = 4 \cdot 8 = 32\).

Key Concepts

Chain Rule

The chain rule is essential in calculus for differentiating composite functions. It allows us to find the derivative of a composition by linking the derivatives of its constituent functions. The chain rule states that if you have two functions, say, \(f(x)\) and \(g(x)\), and you form their composition \(g(f(x))\), the derivative is given by:

• \((g \, \circ \, f)'(x) = g'(f(x)) \, \cdot \, f'(x)\)

This formula helps calculate the rate of change of the entire composition by combining the rates of change of the parts. It effectively 'chains' the derivatives together. The function \(f(x)\) acts as the inner function, while \(g(x)\) is considered the outer function. This method greatly simplifies finding derivatives without explicit differentiation of the entire composite function.

Function Composition

Function composition is a fundamental concept in mathematics. It combines two functions into one to form a new function. Given functions \(f(x)\) and \(g(x)\), their composition \((g \, \circ \, f)(x)\) is defined as:

• \(g(f(x))\)

Here, you input \(x\) into \(f(x)\) and then take the result of \(f(x)\) as the input for \(g(x)\). This operation essentially "nest" two functions within each other, allowing for more complex transformations. Function compositions are useful because they provide a streamlined way to express the sequential application of two operations. In real-world scenarios, it models situations where a process affects another, resulting in a compound effect.

Derivatives

Derivatives are a central concept in calculus, representing the rate at which a function changes with respect to a variable. They provide crucial information on the behavior of functions, such as understanding motion, growth, and rates of change. Mathematically, if you have a function \(f(x)\), its derivative \(f'(x)\) is given by:

• \(\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)

This formula calculates how much \(f(x)\) changes as \(x\) shifts by a small amount \(h\). Hence, the derivative tells you how the output of \(f(x)\) varies at any point \(x\). In graphical terms, the derivative indicates the slope of the tangent to the function's curve at any given point.

Composite Functions

Composite functions combine multiple functions into one. These are often represented by the notation \((g \, \circ \, f)(x)\), which reads as \(g(f(x))\). The composite function considers the output of one function as the input to another, thereby chaining their effects. When dealing with these functions, it's crucial to understand not only the individual functions but their interaction as well.
For example, if \(f(x)\) produces an output \(y\), that output becomes the input to \(g(y)\) when forming \(g(f(x))\).
Composite functions feature prominently in scenarios requiring layered processing, such as signal processing or optimization, where every stage builds on the outcomes of the previous ones.