Question

Finding and Evaluating a Derivative In Exercises \(13-18,\) find \(f^{\prime}(x)\) and \(f^{\prime}(c) .\) $$ \text{Function} \quad \text {Value of} \quad c $$ $$ f(x)=\frac{x-4}{x+4} \quad c=3 $$

Short answer

The derivative of the function \(f(x)\) is \(f^{\prime}(x) = 8/(x+4)^{2}\) and the derivative at \(c=3\) is \(f^{\prime}(3) = 8/49\).

Step-by-step solution

Differentiating the function

Apply quotient rule to find derivative of \(f(x)\). Here \(u=(x-4)\) and \(v=(x+4)\). Now, \((u/v)^{\prime}=(vu'-uv')/v^{2}\). By plugging in the values, we get \(f^{\prime}(x) = \{(x+4)1 - (x-4)1\} / (x+4)^{2} = 8/(x+4)^{2}\). During differentiation, denotative constant 4 was dropped as derivative of constant is 0.

Evaluate the derivative at a specific point

The given point is \(x = c = 3\). Substitute this into the derivative: \( f^{\prime}(3) = 8/(3+4)^{2} = 8/7^2 = 8/49

Key concepts

Quotient Rule

The Quotient Rule is an essential method in calculus for differentiating functions that are expressed as a ratio or fraction. When dealing with a function that has the form \( \frac{u}{v} \), where both \(u\) and \(v\) are differentiable functions, the quotient rule becomes handy. This rule simplifies the process and gives us the formula to find the derivative of such functions.
To apply the quotient rule, the formula we use is:

• \( \left( \frac{u}{v} \right)' = \frac{vu' - uv'}{v^2} \)

Here, \( u' \) is the derivative of the numerator function \(u\), and \(v'\) is the derivative of the denominator function \(v\). The terms \(vu' - uv'\) represent the difference between multiplying \(v\) and the derivative of \(u\), with multiplying \(u\) and the derivative of \(v\).
This formula ensures that we account for both parts of the fraction and their changes in relation to each other. When you become familiar with this rule, it can streamline otherwise complicated differentiation problems involving fractions.

Evaluate Derivative

Once we have found the derivative of a function using the quotient rule, it is often essential to evaluate this derivative at a specific point. Evaluating the derivative allows us to determine the rate of change of the function at that particular point.
For the given problem, after applying the quotient rule, we obtained the derivative:

• \( f'(x) = \frac{8}{(x+4)^2} \)

The task is then to evaluate this at \( x = 3 \). This involves substituting 3 for every instance of \(x\) in the derivative to find the exact rate of change at that point:

• \( f'(3) = \frac{8}{(3+4)^2} = \frac{8}{49} \)

This result tells us how quickly the function \( f(x) \) is changing with respect to \(x\) when \(x\) is equal to 3. It provides practical insights, especially in contexts like physics or engineering, where knowing the rate of change at a specific point is crucial.

Differentiation

Differentiation is the mathematical process of finding a derivative, which is the measure of how a function changes as its input changes. It is a core concept in calculus and serves as the foundation for many real-world applications.
In simple terms, differentiation helps us understand the sensitivity of one variable with respect to changes in another. It provides us the "instantaneous rate of change" or the "slope" of a curve at any point.
For beginners, it is crucial to break down the differentiation process into manageable steps. When applied to simple functions, differentiation can usually be performed using basic rules like the power rule. For more complex functions involving products or quotients, using the product rule or quotient rule becomes imperative.
Fundamentally, differentiation deals with limits and helps create a mathematical representation for rates of change and slopes in graphs. Through practice, mastery of differentiation can open doors to solving diverse problems across physics, engineering, economics, and beyond.