Problem 29 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the higher-order derivative. $$ f^{\prime}(x)=2 x^{2} $$

Step-by-Step Solution

Answer

The second derivative of the function, denoted as \(f''(x)\), is \(4x\).

Step 1: Identify the Existing Function

The existing function is already a derivative, namely the first derivative of some unknown function. It is denoted as \(f'(x) = 2x^{2}\). The task is to find the higher-order derivative, which in this case is the second derivative.

Step 2: Apply the Power Rule

The power rule states that the derivative of \(x^{n}\) is \(n \cdot x^{(n-1)}\). Here, applying the rule to our function \(2x^{2}\), we have: \(2 \cdot 2 \cdot x^{(2-1)} = 4x^{1}\).

Step 3: Write the Final Result

After simplification, the function can be written as \(4x\). This is the second derivative of the original function whose first derivative was given. It is denoted as \(f''(x) = 4x\).

Key Concepts

Power Rule

The power rule is one of the most fundamental rules of differentiation. It comes into play when you need to find derivatives of simple polynomial functions. When you have a function of the form \( x^n \), the power rule tells you that its derivative is \( n \cdot x^{n-1} \).This means you take the exponent of \( x \), multiply it by the coefficient of \( x^n \), and then subtract one from the exponent. Here's a quick breakdown:

• Identify the term \( x^n \).
• Multiply \( n \) by any existing coefficient.
• Subtract one from the exponent \( n \).

Let's take the function \( f(x) = x^3 \) as an example. The power rule tells us that its derivative is \( 3 \cdot x^{3-1} = 3x^2 \). Using this rule makes differentiating polynomial expressions much simpler, as it eliminates more complex calculations. It is a powerful tool to keep in your mathematical toolkit, especially when dealing with higher-order derivatives.

Second Derivative

The second derivative gives us information about the curvature or concavity of the function's graph. It is simply the derivative of the first derivative. In other words, if the first derivative \( f'(x) \) tells you the slope of the tangent to the curve at a particular point, the second derivative \( f''(x) \) tells you how that slope is changing at the same point.Understanding the behavior of the second derivative can help you determine:

• Concavity of a function: If \( f''(x) > 0 \), the function is concave up at that point, resembling a U-shape. If \( f''(x) < 0 \), the function is concave down, resembling an upside-down U.
• Points of inflection: These are points where the concavity changes from up to down, or vice versa. They occur where \( f''(x) = 0 \).

In our exercise, the second derivative of the given function \( f'(x) = 2x^2 \) was found to be \( f''(x) = 4x \). This derivative helps us predict the nature of the curve and its interactions with the x-axis.

Derivative Calculation

Calculating derivatives is a central component of calculus and crucial to understanding how functions behave. A derivative represents the rate of change of a function with respect to its variable, most often \( x \). The steps to calculate the derivative often employ rules like the power rule, product rule, or chain rule, depending on the complexity of the expression.Here's a simple approach to calculating derivatives:

• Identify the function or expression that needs to be differentiated, such as \( f(x) = 2x^2 \).
• Apply the appropriate rule of differentiation to find the first derivative, \( f'(x) \).
• If you're tasked with finding higher-order derivatives, continue differentiating. For example, differentiate \( f'(x) = 2x^2 \) to find the second derivative, resulting in \( f''(x) = 4x \).

The process can be as simple or complex as the function itself, but by applying the right rules systematically, you can always find the derivative you need. Practicing these calculations helps solidify the understanding of how functions change and how their behavior is quantified mathematically.