The Power Rule and the Sum Rule

Oct 13, 2025 by windailys.com 79 views

Introduction

In calculus, the second derivative of a function is a crucial concept that helps us understand the behavior of the function at a given point. It is a measure of how the rate of change of the function is changing. In this article, we will focus on finding the second derivative of the function $y = 7x^3 - 3x^2 + 6$ . We will use the power rule and the sum rule to find the first and second derivatives of the function.

The Power Rule and the Sum Rule

Before we dive into finding the second derivative, let's review the power rule and the sum rule. The power rule states that if $f(x) = x^n$ , then $f'(x) = nx^{n-1}$ . The sum rule states that if $f(x) = g(x) + h(x)$ , then $f'(x) = g'(x) + h'(x)$ .

Finding the First Derivative

To find the first derivative of the function $y = 7x^3 - 3x^2 + 6$ , we will use the power rule and the sum rule. We will differentiate each term separately and then combine the results.

Differentiating the First Term

The first term is $7x^3$ . Using the power rule, we have:

\frac{d}{dx}(7x^3) = 7 \cdot 3x^{3-1} = 21x^2

Differentiating the Second Term

The second term is $-3x^2$ . Using the power rule, we have:

\frac{d}{dx}(-3x^2) = -3 \cdot 2x^{2-1} = -6x

Differentiating the Third Term

The third term is $6$ . Since the derivative of a constant is zero, we have:

\frac{d}{dx}(6) = 0

Combining the Results

Now that we have differentiated each term separately, we can combine the results to find the first derivative of the function:

y' = 21x^2 - 6x

Finding the Second Derivative

To find the second derivative of the function, we will differentiate the first derivative. We will use the power rule and the sum rule again.

Differentiating the First Term

The first term is $21x^2$ . Using the power rule, we have:

\frac{d}{dx}(21x^2) = 21 \cdot 2x^{2-1} = 42x

Differentiating the Second Term

The second term is $-6x$ . Using the power rule, we have:

\frac{d}{dx}(-6x) = -6 \cdot 1x^{1-1} = -6

Combining the Results

Now that we have differentiated each term separately, we can combine the results to find the second derivative of the function:

y'' = 42x - 6

Conclusion

In this article, we found the second derivative of the function $y = 7x^3 - 3x^2 + 6$ . We used the power rule and the sum rule to find the first and second derivatives of the function. The second derivative is a measure of how the rate of change of the function is changing. It is an important concept in calculus that helps us understand the behavior of the function at a given point.

Answer

The correct answer is A) $42x - 6$ .

Discussion

The second derivative of a function is a measure of how the rate of change of the function is changing. It is an important concept in calculus that helps us understand the behavior of the function at a given point. In this article, we found the second derivative of the function $y = 7x^3 - 3x^2 + 6$ . We used the power rule and the sum rule to find the first and second derivatives of the function.

Example Problems

Find the second derivative of the function $y = 2x^4 - 5x^3 + 3x^2 + 1$ .
Find the second derivative of the function $y = x^5 - 2x^4 + 3x^3 - 4x^2 + 1$ .

Solutions

To find the second derivative of the function $y = 2x^4 - 5x^3 + 3x^2 + 1$ , we will differentiate the first derivative. We will use the power rule and the sum rule again.

Differentiating the First Term

The first term is $2x^4$ . Using the power rule, we have:

\frac{d}{dx}(2x^4) = 2 \cdot 4x^{4-1} = 8x^3

Differentiating the Second Term

The second term is $-5x^3$ . Using the power rule, we have:

\frac{d}{dx}(-5x^3) = -5 \cdot 3x^{3-1} = -15x^2

Differentiating the Third Term

The third term is $3x^2$ . Using the power rule, we have:

\frac{d}{dx}(3x^2) = 3 \cdot 2x^{2-1} = 6x

Differentiating the Fourth Term

The fourth term is $1$ . Since the derivative of a constant is zero, we have:

\frac{d}{dx}(1) = 0

Combining the Results

Now that we have differentiated each term separately, we can combine the results to find the first derivative of the function:

y' = 8x^3 - 15x^2 + 6x

Differentiating the First Term

The first term is $8x^3$ . Using the power rule, we have:

\frac{d}{dx}(8x^3) = 8 \cdot 3x^{3-1} = 24x^2

Differentiating the Second Term

The second term is $-15x^2$ . Using the power rule, we have:

\frac{d}{dx}(-15x^2) = -15 \cdot 2x^{2-1} = -30x

Differentiating the Third Term

The third term is $6x$ . Using the power rule, we have:

\frac{d}{dx}(6x) = 6 \cdot 1x^{1-1} = 6

Combining the Results

Now that we have differentiated each term separately, we can combine the results to find the second derivative of the function:

y'' = 24x^2 - 30x + 6

To find the second derivative of the function $y = x^5 - 2x^4 + 3x^3 - 4x^2 + 1$ , we will differentiate the first derivative. We will use the power rule and the sum rule again.

Differentiating the First Term

The first term is $x^5$ . Using the power rule, we have:

\frac{d}{dx}(x^5) = 5x^4

Differentiating the Second Term

The second term is $-2x^4$ . Using the power rule, we have:

\frac{d}{dx}(-2x^4) = -2 \cdot 4x^{4-1} = -8x^3

Differentiating the Third Term

The third term is $3x^3$ . Using the power rule, we have:

\frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2

Differentiating the Fourth Term

The fourth term is $-4x^2$ . Using the power rule, we have:

\frac{d}{dx}(-4x^2) = -4 \cdot 2x^{2-1} = -8x

Differentiating the Fifth Term

The fifth term is $1$ . Since the derivative of a constant is zero, we have:

\frac{d}{dx}(1) = 0

Combining the Results

Now that we have differentiated each term separately, we can combine the results to find the first derivative of the function:

y' = 5x^4 - 8x^3 + 9x^2 - 8x

Differentiating the First Term

The first term is $5x^4$ . Using the power rule, we have:

\frac{d}{dx}(5x^4) = 5 \cdot 4x^{4-1} = 20x^3

Differentiating the Second Term

The second term is $-8x^3$ . Using the power rule, we have:

\frac{d}{dx}(-8x^3) = -8 \cdot 3x^{3-1} = -24x^2

Differentiating the Third Term

The third term is $9x^2$ . Using the power rule, we have:

\frac{d}{dx}(9x^2)<br/> # **Q&A: Finding the Second Derivative of a Function** ## **Introduction** In our previous article, we discussed how to find the second derivative of a function using the power rule and the sum rule. We used the function $y = 7x^3 - 3x^2 + 6$ as an example and found the second derivative to be $y'' = 42x - 6$. In this article, we will answer some frequently asked questions about finding the second derivative of a function. ## **Q: What is the second derivative of a function?** A: The second derivative of a function is a measure of how the rate of change of the function is changing. It is an important concept in calculus that helps us understand the behavior of the function at a given point. ## **Q: How do I find the second derivative of a function?** A: To find the second derivative of a function, you need to differentiate the first derivative of the function. You can use the power rule and the sum rule to find the first derivative, and then differentiate the result to find the second derivative. ## **Q: What is the power rule?** A: The power rule is a rule in calculus that states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. This rule is used to find the derivative of a function that is raised to a power. ## **Q: What is the sum rule?** A: The sum rule is a rule in calculus that states that if $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$. This rule is used to find the derivative of a function that is the sum of two or more functions. ## **Q: How do I use the power rule and the sum rule to find the second derivative of a function?** A: To use the power rule and the sum rule to find the second derivative of a function, you need to follow these steps: 1. Find the first derivative of the function using the power rule and the sum rule. 2. Differentiate the first derivative to find the second derivative. 3. Use the power rule and the sum rule to simplify the result. ## **Q: What are some common mistakes to avoid when finding the second derivative of a function?** A: Some common mistakes to avoid when finding the second derivative of a function include: * Forgetting to use the power rule and the sum rule. * Not differentiating the first derivative correctly. * Not simplifying the result correctly. ## **Q: How do I check my work when finding the second derivative of a function?** A: To check your work when finding the second derivative of a function, you can use the following steps: 1. Check that you have used the power rule and the sum rule correctly. 2. Check that you have differentiated the first derivative correctly. 3. Check that you have simplified the result correctly. ## **Q: What are some real-world applications of the second derivative of a function?** A: The second derivative of a function has many real-world applications, including: * Physics: The second derivative of a function is used to describe the acceleration of an object. * Engineering: The second derivative of a function is used to design and optimize systems. * Economics: The second derivative of a function is used to model and analyze economic systems. ## **Conclusion** In this article, we have answered some frequently asked questions about finding the second derivative of a function. We have discussed the power rule and the sum rule, and how to use them to find the second derivative of a function. We have also discussed some common mistakes to avoid and how to check your work. Finally, we have discussed some real-world applications of the second derivative of a function. ## **Example Problems** 1. Find the second derivative of the function $y = 2x^4 - 5x^3 + 3x^2 + 1$. 2. Find the second derivative of the function $y = x^5 - 2x^4 + 3x^3 - 4x^2 + 1$. ## **Solutions** 1. To find the second derivative of the function $y = 2x^4 - 5x^3 + 3x^2 + 1$, we will differentiate the first derivative. We will use the power rule and the sum rule again. ### **Differentiating the First Term** The first term is $2x^4$. Using the power rule, we have: $$\frac{d}{dx}(2x^4) = 2 \cdot 4x^{4-1} = 8x^3

Differentiating the Second Term

The second term is $-5x^3$ . Using the power rule, we have:

\frac{d}{dx}(-5x^3) = -5 \cdot 3x^{3-1} = -15x^2

Differentiating the Third Term

The third term is $3x^2$ . Using the power rule, we have:

\frac{d}{dx}(3x^2) = 3 \cdot 2x^{2-1} = 6x

Differentiating the Fourth Term

The fourth term is $1$ . Since the derivative of a constant is zero, we have:

\frac{d}{dx}(1) = 0

Combining the Results

Now that we have differentiated each term separately, we can combine the results to find the first derivative of the function:

y' = 8x^3 - 15x^2 + 6x

Differentiating the First Term

The first term is $8x^3$ . Using the power rule, we have:

\frac{d}{dx}(8x^3) = 8 \cdot 3x^{3-1} = 24x^2

Differentiating the Second Term

The second term is $-15x^2$ . Using the power rule, we have:

\frac{d}{dx}(-15x^2) = -15 \cdot 2x^{2-1} = -30x

Differentiating the Third Term

The third term is $6x$ . Using the power rule, we have:

\frac{d}{dx}(6x) = 6 \cdot 1x^{1-1} = 6

Combining the Results

Now that we have differentiated each term separately, we can combine the results to find the second derivative of the function:

y'' = 24x^2 - 30x + 6

To find the second derivative of the function $y = x^5 - 2x^4 + 3x^3 - 4x^2 + 1$ , we will differentiate the first derivative. We will use the power rule and the sum rule again.

Differentiating the First Term

The first term is $x^5$ . Using the power rule, we have:

\frac{d}{dx}(x^5) = 5x^4

Differentiating the Second Term

The second term is $-2x^4$ . Using the power rule, we have:

\frac{d}{dx}(-2x^4) = -2 \cdot 4x^{4-1} = -8x^3

Differentiating the Third Term

The third term is $3x^3$ . Using the power rule, we have:

\frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2

Differentiating the Fourth Term

The fourth term is $-4x^2$ . Using the power rule, we have:

\frac{d}{dx}(-4x^2) = -4 \cdot 2x^{2-1} = -8x

Differentiating the Fifth Term

The fifth term is $1$ . Since the derivative of a constant is zero, we have:

\frac{d}{dx}(1) = 0

Combining the Results

Now that we have differentiated each term separately, we can combine the results to find the first derivative of the function:

y' = 5x^4 - 8x^3 + 9x^2 - 8x

Differentiating the First Term

The first term is $5x^4$ . Using the power rule, we have:

\frac{d}{dx}(5x^4) = 5 \cdot 4x^{4-1} = 20x^3

Differentiating the Second Term

The second term is $-8x^3$ . Using the power rule, we have:

\frac{d}{dx}(-8x^3) = -8 \cdot 3x^{3-1} = -24x^2

Differentiating the Third Term

The third term is $9x^2$ . Using the power rule, we have:

\frac{d}{dx}(9x^2) = 9 \cdot 2x^{2-1} = 18x

Differentiating the Fourth Term

The fourth term is $-8x$ . Using the power rule, we have:

\frac{d}{dx}(-8x) = -8 \cdot 1x^{1-1} = -8

Combining the Results

Now that we have differentiated each