Problem 1 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the second derivative of the function. $$ f(x)=9-2 x $$

Step-by-Step Solution

Answer

The second derivative of the function \( f(x)= 9-2x \) is 0.

Step 1: Differentiate the function

The derivative of the function \( f(x)= 9-2x \) is obtained by applying the power rule of differentiation, which states that the derivative of \( x^n \) is \( n*x^{n-1} \). The derivative of \( f(x) \) is, therefore: \[ f'(x)= -2 \]

Step 2: Differentiate again

The second derivative is the derivative of the first derivative. Since the first derivative is a constant, the derivative of a constant is 0. So the second derivative \( f''(x) \) is: \[ f''(x)= 0 \]

Key Concepts

Power Rule of Differentiation

Understanding the power rule of differentiation is crucial when tackling calculus problems involving derivatives. It's a method used to find the derivative of a function that is a power of x. The basic formula says that if you have a function in the form of \( f(x) = x^n \), where \( n \) is any real number, the derivative of that function, denoted as \( f'(x) \), is \( nx^{n-1} \).

For example, if you have a function like \( g(x) = x^3 \), using the power rule, you'd calculate the derivative, \( g'(x) \), as \( 3x^{3-1} = 3x^2 \). The power rule greatly simplifies the process of differentiation by providing a quick and reliable method for finding derivatives of polynomial terms.

Derivative of a Constant

In calculus, the derivative of a constant is a foundational concept that pops up frequently in various calculations. A constant, as the name suggests, doesn't change. It has no dependence on variables like x or y. Because the derivative measures how a function changes as its input changes, and a constant never changes regardless of the input, the derivative of any constant is zero.

This is why in the step-by-step solution for the function \( f(x) = 9 - 2x \), the derivative of the constant term 9 was not explicitly mentioned; it disappears in the differentiation process, leaving us only with the derivative of the linear term \( -2x \). This simplification plays a key role in solving calculus problems efficiently and accurately.

Calculus Problem Solving

Calculus problem solving often involves finding derivatives and understanding the nature of the function based on those derivatives. The first derivative provides insights into the function's rate of change, or in other words, its slope. For instance, a positive derivative indicates an increasing function, whereas a negative derivative shows that the function is decreasing.

The second derivative, which in the given exercise is zero, tells us about the concavity of the function - if it's concave up or down, or in this specific case with a second derivative of zero, the function is linear. Thus, both the power rule of differentiation and the concept of the derivative of a constant are integral to solving calculus problems, as they help to analyze the behavior of functions.