Problem 35 - Brief Applied Calculus

Question

31-38. Find the indicated derivatives. If \(f(x)=x^{3}\), find \(\left.\frac{d f}{d x}\right|_{x=-3}\)

Step-by-Step Solution

Answer

The derivative at \( x = -3 \) is 27.

Step 1: Understand the Problem

We are given a function \( f(x) = x^3 \) and need to find its derivative, then evaluate that derivative at \( x = -3 \).

Step 2: Differentiate the Function

Apply the power rule for derivatives. The power rule states that if \( f(x) = x^n \), then \( \frac{d}{dx}f(x) = nx^{n-1} \). For \( f(x) = x^3 \), the derivative is \( f'(x) = 3x^2 \).

Step 3: Evaluate the Derivative at x = -3

Substitute \( x = -3 \) into the derivative we found. So, calculate \( f'(-3) = 3(-3)^2 = 3 \times 9 = 27 \).

Key Concepts

Power Rule

The power rule is one of the simplest and most commonly used rules in calculus for finding derivatives of polynomial functions. When you have a function of the form

• \(f(x) = x^n\)

the power rule allows you to quickly determine its derivative. The derivative, using the power rule, is:

• \( \frac{d}{dx} x^n = nx^{n-1} \)

This formula is intuitive yet powerful, giving you the slope of the tangent line to the curve at any point.

For instance, consider the function \(f(x) = x^3\). Here, \(n\) equals 3. Applying the power rule:

• Bring down the exponent (3) as a coefficient.
• Subtract one from the exponent.

So, the derivative becomes \(f'(x) = 3x^{2}\). It's important to practice using the power rule, as this will give you confidence to handle different powers and polynomial expressions efficiently.

Evaluating Derivatives

Once you have determined the derivative of a function, the next step is often to evaluate it at a specific point. Evaluating a derivative means finding the slope of the function at this particular point, which is crucial in understanding how the function behaves locally.

Let's say you have found the derivative of the function \( f(x) = x^3 \), which is \( f'(x) = 3x^2 \). To evaluate this derivative at \( x = -3 \), you substitute \( -3 \) into the derivative:

• \( f'(-3) = 3(-3)^2 \)
• \( = 3 \cdot 9 \)
• \( = 27 \)

Therefore, the slope at \( x = -3 \) is 27. This tells you that at \( x = -3 \), the function is increasing quite steeply. Understanding how to evaluate derivatives is essential for analyzing real-world problems and determining rates of change.

Function Differentiation

Differentiation is the process of finding the derivative of a function. This concept is fundamental in calculus as it involves calculating the rate at which a function is changing at any given point. Differentiation is not just about applying rules; it’s about understanding how a function behaves.

To differentiate a function completely:

• Identify the form of the function and choose the appropriate rule (like the power rule, product rule, or chain rule)
• Apply the selected rule to find the derivative

Take \( f(x) = x^3 \) as an example. Recognize it as a power function, then apply the power rule:

• \( f'(x) = 3x^2 \)

This simple differentiation tells you how the function’s slope changes and is particularly useful for modeling and prediction. Differentiation underpins much of the analysis in physics, engineering, and economics by enabling you to see the effects of variable change.