Problem 71 - 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 \(\quad h^{\prime}(2)=4\) $$ f(x)=g(x)+h(x) $$

Step-by-Step Solution

Answer

The value of \(f'(2)\) is 2.

Step 1: Understand the problem

The function \(f(x)\) is the sum of \(g(x)\) and \(h(x)\), so \(f'(x)\) will be the sum of \(g'(x)\) and \(h'(x)\). This is derived from the sum rule in differentiation.

Step 2: Apply the sum rule

According to the sum rule, \(f'(x) = g'(x) + h'(x)\). We are given the values of \(g'(x)\) and \(h'(x)\) at x=2. So, \(f'(2) = g'(2) + h'(2)\).

Step 3: substitute the given values

Substitute the given values into the equation from step 2. \(f'(2) = g'(2) + h'(2) = -2 + 4 = 2.\)

Key Concepts

Sum Rule

The Sum Rule in differentiation is a fundamental concept that makes calculating derivatives of functions much simpler. It states that if you have two differentiable functions, say \( g(x) \) and \( h(x) \), their sum \( f(x) = g(x) + h(x) \) is also differentiable. For the derivative of this sum, the Sum Rule tells us that:

• \( f'(x) = g'(x) + h'(x) \)

This means the derivative of the entire function \( f(x) \) is simply the sum of the derivatives of \( g(x) \) and \( h(x) \).

To apply this, you do not need to reinvent the wheel for every problem. Instead, check that both functions are differentiable, and then add their derivatives to get the derivative of their sum.In our original exercise, we have \( f(x) = g(x) + h(x) \), where \( f'(x) \) was found by adding \( g'(x) \) and \( h'(x) \). This shows how efficiently the Sum Rule can be applied to find the derivative of combined functions.

Derivative of a Function

The derivative of a function is essentially a measure of how a function's output changes as its input changes. More formally, if you have a function \( f(x) \), its derivative, denoted as \( f'(x) \), represents the rate of change or slope of \( f \) at any given point.
Here's a simple way to think about derivatives:

• They give you the instantaneous rate of change at any point on the function.
• They're used to find tangent lines, which tell the slope at a specific point.
• They help in understanding the behavior of the function, such as increasing or decreasing trends.

In the context of the exercise, understanding derivatives helps us know how the function \( f(x) = g(x) + h(x) \) changes around \( x = 2 \). By finding \( f'(2) \), we gain insights into the behavior of \( f \) at that specific point.

Evaluation of Derivatives

Evaluation of derivatives is the process of finding the derivative of a function at a particular point. In our exercise, we needed to evaluate the derivative \( f'(x) \) at \( x = 2 \), using the information given for \( g'(x) \) and \( h'(x) \) at this point.
Here's how you can evaluate derivatives effectively:

• Substitute the given values for derivatives into the derivative expression.
• For our function \( f'(2) = g'(2) + h'(2) \), substitute \( g'(2) = -2 \) and \( h'(2) = 4 \).
• Simply calculate the sum of these given values:

You carry out the math: \[f'(2) = -2 + 4 = 2\]This completion of our evaluation shows how the given derivative values impact the function \( f \) at a particular point, allowing us to understand the local behavior of the sum of the functions.