Problem 72 - 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 \(\quad h^{\prime}(2)=4\) $$ f(x)=\frac{g(x)}{h(x)} $$

Step-by-Step Solution

Answer

The value of the derivative of the function \(f(x)\) at \(x=2\), \(f^{\prime}(2)\), is -10.

Step 1: Apply the Quotient Rule

Firstly, apply the Quotient Rule to the original function \(f(x)\), so \(f^{\prime}(x) = \frac{g^{\prime}(x)h(x) - g(x)h^{\prime}(x)}{(h(x))^2}\)

Step 2: Substitute \(x=2\) into \(f^{\prime}(x)\)

Next, substitute \(x=2\) into the derivative function: \(f^{\prime}(2) = \frac{g^{\prime}(2)h(2) - g(2)h^{\prime}(2)}{(h(2))^2}\)

Step 3: Substitute Values

The next task is to substitute the given function values into the equation: \(f^{\prime}(2) = \frac{(-2)(-1) - 3*4}{(-1)^2}\)

Step 4: Simplify the Equation

The last step is to simplify the equation to find the final derivative value: \(f^{\prime}(2) = \frac{2 - 12}{1} = -10

Key Concepts

Quotient Rule

Learning how to differentiate is a fundamental skill in calculus, and the Quotient Rule is one of the essential tools used for function differentiation. When dealing with the differentiation of functions presented as a ratio of two differentiable functions, the Quotient Rule is your method of choice. The Quotient Rule states that for two differentiable functions, g(x) and h(x), the derivative of their quotient is given by:
\[ f^{\text{'}}(x) = \frac{g^{\text{'}}(x)h(x) - g(x)h^{\text{'}}(x)}{(h(x))^2} \]
This elegant formula allows us to handle complex fractions without having to perform long division or other more laborious methods to simplify them first. Instead, we can directly apply this formula to find the derivative of the quotient quickly and accurately. Remember to apply it step-by-step to avoid mistakes, and verify your solution by confirming that all components of the formula—g(x), g'(x), h(x), and h'(x)—are correctly placed and computed.

Derivative

The concept of a derivative is the cornerstone of calculus. It represents the rate at which a function is changing at any given point. For a function f(x), the derivative f'(x) can be interpreted as the slope of the tangent line to the function's graph at x. The process of finding this derivative is called differentiation.
One can picture the derivative as providing a snapshot of the function's behavior—whether it's increasing, decreasing, or stagnant—exactly at the point of interest. The applications of derivatives are vast, including but not limited to physics for velocity and acceleration calculations, in economics for understanding marginal cost and revenue, or in biology for modeling population growth. The derivative is truly an essential concept that underpins many advanced theories and real-world applications.

Function Differentiation

Function differentiation is the action of applying calculus methods to find the derivative of a function. The objective is to understand how the output of the function changes with respect to a change in the input. To achieve this, we use different rules of differentiation depending on the structure of the function, such as product rule, chain rule, power rule, and, as in the exercise provided, the Quotient Rule.

When learning function differentiation, it's important to first grasp the idea of the derivative intuitively and then to get comfortable with the various rules that can be applied to different functions. Each rule has specific cases where it applies, and mastering these rules will allow you to tackle a wide range of problems, from simple polynomials to complex rational functions.

Calculus

Calculus is a vast field of mathematics that studies continuous change, encompassing techniques and theories such as derivatives, integrals, limits, and infinite series. It's divided into two main branches: differential calculus, which focuses on the rate of change and slopes of curves; and integral calculus, which deals with accumulation of quantities and areas under or between curves.

Through calculus, we can solve problems that involve calculating the area of an irregular shape, the growth rate of a tumor, or the changing conditions of a moving vehicle, among many others. Its development can be attributed to great mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz, and its applications are seen in every science, engineering, and economic field. Understanding the basics of calculus, such as derivatives and the Quotient Rule, is an essential step for anyone looking to delve into the sciences or related quantitative fields.