Find $f^{\prime \prime}(x)$.$f(x) = 5x^2 - 13x - \frac{8}{x^3}$$ F ′ ′ ( X ) = F^{\prime \prime}(x) = F ′′ ( X ) = [/tex] $\square$

Introduction

In calculus, derivatives and second derivatives are fundamental concepts used to study the behavior of functions. The first derivative, denoted as $f^{\prime}(x)$, represents the rate of change of a function with respect to its input variable. The second derivative, denoted as $f^{\prime \prime}(x)$, represents the rate of change of the first derivative with respect to the input variable. In this article, we will focus on finding the second derivative of a given function.

The Given Function

The given function is $f(x) = 5x^2 - 13x - \frac{8}{x^3}$. To find the second derivative, we need to first find the first derivative of this function.

Finding the First Derivative

To find the first derivative of the given function, we will apply the power rule of differentiation, which states that if $f(x) = x^n$, then $f^{\prime}(x) = nx^{n-1}$. We will also use the quotient rule of differentiation, which states that if $f(x) = \frac{g(x)}{h(x)}$, then $f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{[h(x)]^2}$.

Using the power rule, we can find the derivative of the first two terms of the given function:

$$\frac{d}{dx}(5x^2) = 10x$$

$$\frac{d}{dx}(-13x) = -13$$

To find the derivative of the third term, we will use the quotient rule:

$$\frac{d}{dx}\left(-\frac{8}{x^3}\right) = \frac{x^3(-8)^{\prime} - (-8)(x^3)^{\prime}}{(x^3)^2}$$

$$= \frac{x^3(0) + 8(3x^2)}{(x^3)^2}$$

$$= \frac{24x^2}{x^6}$$

$$= \frac{24}{x^4}$$

Now, we can combine the derivatives of the three terms to find the first derivative of the given function:

$$f^{\prime}(x) = 10x - 13 + \frac{24}{x^4}$$

Finding the Second Derivative

To find the second derivative of the given function, we need to differentiate the first derivative with respect to the input variable. We will apply the power rule of differentiation again:

$$\frac{d}{dx}(10x) = 10$$

$$\frac{d}{dx}(-13) = 0$$

To find the derivative of the third term, we will use the power rule:

$$\frac{d}{dx}\left(\frac{24}{x^4}\right) = \frac{24}{x^4}(-4x^{-5})$$

$$= -\frac{96}{x^9}$$

Now, we can combine the derivatives of the three terms to find the second derivative of the given function:

$$f^{\prime \prime}(x) = 10 - \frac{96}{x^9}$$

Conclusion

In this article, we have found the second derivative of a given function using the power rule and the quotient rule of differentiation. The second derivative represents the rate of change of the first derivative with respect to the input variable. We have shown that the second derivative of the given function is $f^{\prime \prime}(x) = 10 - \frac{96}{x^9}$.

Applications of Second Derivatives

Second derivatives have numerous applications in various fields, including physics, engineering, and economics. Some of the key applications of second derivatives include:

• Optimization: Second derivatives are used to find the maximum or minimum of a function.
• Motion: Second derivatives are used to describe the acceleration of an object.
• Economics: Second derivatives are used to study the behavior of economic systems.

Real-World Examples

1. Projectile Motion: The second derivative of the position function of a projectile is used to describe its acceleration.
2. Economic Systems: The second derivative of the production function of an economic system is used to study its behavior.
3. Optimization: The second derivative of a function is used to find its maximum or minimum value.

Conclusion

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Introduction

In our previous article, we discussed the concept of second derivatives and their applications in various fields. In this article, we will answer some frequently asked questions about second derivatives and their uses.

Q: What is the difference between a first derivative and a second derivative?

A: The first derivative of a function represents the rate of change of the function with respect to its input variable. The second derivative of a function represents the rate of change of the first derivative with respect to the input variable.

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 with respect to the input variable. You can use the power rule and the quotient rule of differentiation to find the second derivative.

Q: What are some common applications of second derivatives?

A: Second derivatives have numerous applications in various fields, including physics, engineering, and economics. Some of the key applications of second derivatives include:

• Optimization: Second derivatives are used to find the maximum or minimum of a function.
• Motion: Second derivatives are used to describe the acceleration of an object.
• Economics: Second derivatives are used to study the behavior of economic systems.

Q: Can you provide some real-world examples of the use of second derivatives?

A: Yes, here are some real-world examples of the use of second derivatives:

1. Projectile Motion: The second derivative of the position function of a projectile is used to describe its acceleration.
2. Economic Systems: The second derivative of the production function of an economic system is used to study its behavior.
3. Optimization: The second derivative of a function is used to find its maximum or minimum value.

Q: How do I use second derivatives to optimize a function?

A: To use second derivatives to optimize a function, you need to find the critical points of the function by setting the first derivative equal to zero. Then, you need to use the second derivative to determine whether the critical point is a maximum or minimum.

Q: What are some common mistakes to avoid when working with second derivatives?

A: Some common mistakes to avoid when working with second derivatives include:

• Not checking the domain of the function: Make sure to check the domain of the function before finding its second derivative.
• Not using the correct rules of differentiation: Make sure to use the correct rules of differentiation, such as the power rule and the quotient rule.
• Not checking the sign of the second derivative: Make sure to check the sign of the second derivative to determine whether the critical point is a maximum or minimum.

Q: Can you provide some tips for working with second derivatives?

A: Yes, here are some tips for working with second derivatives:

• Use a calculator or computer algebra system: Use a calculator or computer algebra system to help you find the second derivative of a function.
• Check your work: Make sure to check your work by plugging in values and checking the results.
• Use the correct notation: Use the correct notation, such as $f^{\prime \prime}(x)$, to represent the second derivative of a function.

Conclusion

In conclusion, second derivatives are a fundamental concept in calculus that have numerous applications in various fields. We have answered some frequently asked questions about second derivatives and their uses, and provided some tips for working with second derivatives. We hope that this article has been helpful in understanding the concept of second derivatives and their applications.