Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Use the given information to find \(f^{\prime}(2)\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=3-g(x) $$

Step-by-Step Solution

Answer

The derivative \(f^{\prime}(2) = 2\).

Step 1: Find the Derivative of \(f(x)\)

First, differentiate \(f(x) =3 - g(x)\) to obtain \(f^{\prime}(x)=- g^{\prime}(x)\). This is because the derivative of a constant is zero and the negative sign arises from the differentiation of \(-g(x)\).

Step 2: Substitute \(x = 2\)

Substitute \(x = 2\) into the derivative of \(f(x)\) to get \(f^{\prime}(2) = - g^{\prime}(2)\). This means that \(f^{\prime}(2)\) is equal to the negative of \(g^{\prime}(2)\).

Step 3: Substitute the given value of \(g^{\prime}(2)\)

Substitute \(g^{\prime}(2) = -2\) into \(f^{\prime}(2) = - g^{\prime}(2)\) to get \(f^{\prime}(2) = 2\).

Key Concepts

Functions

In mathematics, functions are a fundamental concept that describes a relationship between two sets of elements, typically referred to as the domain and the range. Imagine a function as a machine that takes an input, performs a specific operation on it, and then provides an output. The input element comes from the domain, while the output belongs to the range.

Functions are commonly written as \( f(x) \), where \( f \) is the function name, and \( x \) is the variable representing the input. In the given exercise, we are dealing with a function \( f(x) = 3 - g(x) \). Here, \( g(x) \) is another function, meaning \( g \) itself also transforms an input into an output.

Understanding functions is vital as they allow us to model real-world scenarios, solve problems, and perform mathematical operations like differentiation and integration. They establish a connection between different quantities and help simplify complex mathematical expressions. When working with functions, we visualize how changes in input affect the output, providing insights into various phenomena.

Differentiation

Differentiation is a foundational concept in calculus, focusing on finding the rate at which a function changes. When you differentiate a function, you calculate its derivative, which signifies the function's instantaneous rate of change or slope at any given point.

In the context of our exercise, differentiation is used to find the derivative of the function \( f(x) = 3 - g(x) \). By differentiating, we obtain \( f^{\prime}(x) = -g^{\prime}(x) \). Differentiation helps us understand how one variable changes concerning another—critical for analyzing curves, optimization problems, and rate-related scenarios.

To perform differentiation, we apply specific rules, such as the power rule, product rule, and chain rule. These rules guide us in systematically breaking down complex functions into simpler parts for easy calculation of derivatives.

Chain Rule

The chain rule is an essential tool in calculus for differentiating composite functions. A composite function is one where the function is nested within another, like \( h(g(x)) \). The chain rule allows us to differentiate these functions efficiently by linking the derivatives of the inner and outer functions.

In simpler terms, if we have a function \( y = f(g(x)) \), the chain rule states that the derivative \( \frac{dy}{dx} \) is given by \( f'(g(x)) \cdot g'(x) \). This means you first differentiate the outer function at the inner function and then multiply it by the derivative of the inner function.

The chain rule is crucial because many real-world functions can be composite, requiring this rule for accurate differentiation. While our current exercise does not specifically employ the chain rule, understanding it broadens your calculus skills and prepares you for more advanced topics where multiple layers of functions need differentiation. It's crucial for studying rates of change in more complex models.