Problem 127 - Calculus Volume 1

Question

For the following exercises, assume that \(f(x)\) and \(g(x)\) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives. $$\begin{array}{|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} \\ \hline f(x) & {3} & {5} & {-2} & {0} \\ \hline g(x) & {2} & {3} & {-4} & {6} \\ \hline f^{\prime}(x) & {-1} & {7} & {8} & {-3} \\ \hline g^{\prime}(x) & {4} & {1} & {2} & {9} \\ \hline\end{array}$$ Find \(h^{\prime}(2)\) if \(h(x)=\frac{f(x)}{g(x)}\)

Step-by-Step Solution

Answer

\(h'(2) = \frac{16}{9}\).

Step 1: Identify the Derivative Rule

The function given is a quotient, \(h(x) = \frac{f(x)}{g(x)}\). To find the derivative \(h'(x)\), we use the quotient rule, which states: \( h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\).

Step 2: Apply the Quotient Rule

Substitute values from the table into the quotient rule. At \(x = 2\), we have: \(f(x) = 5\), \(g(x) = 3\), \(f'(x) = 7\), and \(g'(x) = 1\). The derivative is \(h'(2) = \frac{7 \cdot 3 - 5 \cdot 1}{3^2}\).

Step 3: Perform the Calculations

Now compute each part: \(7 \cdot 3 = 21\) and \(5 \cdot 1 = 5\). Substituting back gives: \(h'(2) = \frac{21 - 5}{9} = \frac{16}{9}\).

Step 4: Simplify the Expression

The expression \(\frac{16}{9}\) is already in its simplest form, so \(h'(2) = \frac{16}{9}\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function changes at any given point. It allows us to compute the derivative, which represents this rate of change. In simple terms, differentiation tells us how a function's output changes as its input changes by an infinitely small amount.

When applying differentiation, we often look to specific rules and techniques, as not all functions are as straightforward as simple linear equations. Whether dealing with polynomials, products, or quotients like in our current problem, understanding which differentiation rule to apply is crucial.

Quotient Rule is one of those rules specifically designed for functions that are fractions, such as our given function \(h(x) = \frac{f(x)}{g(x)}\). Each function type requires specific attention to detail and careful application of rules to ensure the derivative is calculated correctly.

Function Derivatives

Function derivatives play a key role in understanding how a function behaves, reflecting its slope at any given point. To solve calculus problems involving derivatives, it's important to have a good grasp of the basic derivative rules.

For example, rules like the Product Rule, Chain Rule, and Quotient Rule are vital. In this problem, we focus on the Quotient Rule because of the division between two functions: \(f(x)\) and \(g(x)\). According to the Quotient Rule, the derivative \(h'(x)\) of the function \(h(x) = \frac{f(x)}{g(x)}\) is calculated as:

\[ h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]

Understanding and correctly applying this formula is crucial. Not only does it give us \(h'(2)\) in the context of our exercise, but it's also a tool you'll use often in calculus.

Calculus Problem Solving

Solving calculus problems can be straightforward if you follow a strategic approach. This involves analyzing the problem to determine what rules apply, substituting known values, performing calculations, and simplifying your results.

Let's consider our problem: finding \(h'(2)\) for \(h(x)=\frac{f(x)}{g(x)}\). We begin by correctly identifying that the Quotient Rule is needed. From there, we extract the necessary values from the table:

• \(f(2) = 5\)
• \(g(2) = 3\)
• \(f'(2) = 7\)
• \(g'(2) = 1\)

Next, we apply these to the quotient rule formula to perform the mathematical operations: \(7 \cdot 3 - 5 \cdot 1 = 21 - 5\), resulting in 16. Finally, we divide by \(3^2\), giving us \(\frac{16}{9}\).

By following these steps methodically, calculus problems can become manageable, helping students foster a better understanding and application of mathematical concepts.