Find { F^{\prime}(x) $}$ For The Following Function. Then Find { F^{\prime}(3), F^{\prime}(0) $}$, And { F^{\prime}(-2) $} . . . { F(x) = \frac{-9}{x} \} ${ F^{\prime}(x) = }$

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on finding the derivative of a given function and then use it to calculate the derivative at specific points.

The Given Function

The given function is $f(x) = \frac{-9}{x}$. This is a rational function, and we will use the quotient rule to find its derivative.

Finding the Derivative

The quotient rule states that if we have a function of the form $f(x) = \frac{g(x)}{h(x)}$, then the derivative of $f(x)$ is given by:

$$f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{[h(x)]^2}$$

In our case, $g(x) = -9$ and $h(x) = x$. Therefore, we have:

$$g^{\prime}(x) = 0 \quad \text{and} \quad h^{\prime}(x) = 1$$

Substituting these values into the quotient rule formula, we get:

$$f^{\prime}(x) = \frac{x(0) - (-9)(1)}{x^2} = \frac{9}{x^2}$$

Finding the Derivative at Specific Points

Now that we have found the derivative of the function, we can use it to calculate the derivative at specific points. We are asked to find $f^{\prime}(3)$, $f^{\prime}(0)$, and $f^{\prime}(-2)$.

To find $f^{\prime}(3)$, we substitute $x = 3$ into the derivative formula:

$$f^{\prime}(3) = \frac{9}{3^2} = \frac{9}{9} = 1$$

To find $f^{\prime}(0)$, we substitute $x = 0$ into the derivative formula. However, we must be careful when dealing with division by zero. In this case, the derivative is undefined at $x = 0$.

To find $f^{\prime}(-2)$, we substitute $x = -2$ into the derivative formula:

$$f^{\prime}(-2) = \frac{9}{(-2)^2} = \frac{9}{4}$$

Conclusion

In this article, we found the derivative of the function $f(x) = \frac{-9}{x}$ using the quotient rule. We then used the derivative to calculate the derivative at specific points, including $f^{\prime}(3)$, $f^{\prime}(0)$, and $f^{\prime}(-2)$. We hope that this article has provided a clear and concise explanation of how to find derivatives and has helped you to understand this important concept in mathematics.

Derivative Formula

The derivative of the function $f(x) = \frac{-9}{x}$ is given by:

$$f^{\prime}(x) = \frac{9}{x^2}$$

Derivative at Specific Points

The derivative at specific points is given by:

• $f^{\prime}(3) = 1$
• $f^{\prime}(0)$ is undefined
• $f^{\prime}(-2) = \frac{9}{4}$

Applications of Derivatives

Derivatives have numerous applications in various fields, including:

• Physics: Derivatives are used to describe the motion of objects and to calculate forces and energies.
• Engineering: Derivatives are used to design and optimize systems, such as bridges and buildings.
• Economics: Derivatives are used to model economic systems and to make predictions about future economic trends.

Conclusion

Introduction

In our previous article, we discussed how to find the derivative of a function using the quotient rule. We also calculated the derivative at specific points. In this article, we will answer some frequently asked questions about derivatives.

Q: What is a derivative?

A: A derivative is a measure of how a function changes as its input changes. It represents the rate of change of the function with respect to its input.

Q: Why are derivatives important?

A: Derivatives are important because they help us understand how functions change and behave. They are used in many fields, including physics, engineering, and economics, to model and analyze complex systems.

Q: How do I find the derivative of a function?

A: To find the derivative of a function, you can use various rules, such as the power rule, the product rule, and the quotient rule. The power rule states that if $f(x) = x^n$, then $f^{\prime}(x) = nx^{n-1}$. The product rule states that if $f(x) = g(x)h(x)$, then $f^{\prime}(x) = g^{\prime}(x)h(x) + g(x)h^{\prime}(x)$. The quotient rule states that if $f(x) = \frac{g(x)}{h(x)}$, then $f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{[h(x)]^2}$.

Q: What is the difference between a derivative and a differential?

A: A derivative is a measure of how a function changes as its input changes, while a differential is a measure of the change in the function itself. In other words, the derivative tells us how fast the function is changing, while the differential tells us the actual change in the function.

Q: Can I use derivatives to find the maximum or minimum of a function?

A: Yes, you can use derivatives to find the maximum or minimum of a function. If the derivative of a function is zero at a point, then that point is a critical point, and the function may have a maximum or minimum at that point. To determine whether the point is a maximum or minimum, you can use the second derivative test.

Q: How do I use the second derivative test to determine whether a point is a maximum or minimum?

A: To use the second derivative test, you need to find the second derivative of the function. If the second derivative is positive at a point, then the function has a minimum at that point. If the second derivative is negative at a point, then the function has a maximum at that point. If the second derivative is zero at a point, then the test is inconclusive.

Q: Can I use derivatives to model real-world phenomena?

A: Yes, you can use derivatives to model real-world phenomena. Derivatives are used in many fields, including physics, engineering, and economics, to model and analyze complex systems. For example, you can use derivatives to model the motion of objects, the growth of populations, and the behavior of financial markets.

Conclusion

In conclusion, derivatives are a powerful tool for understanding how functions change and behave. They are used in many fields to model and analyze complex systems. We hope that this article has provided a clear and concise explanation of derivatives and has helped you to understand this important concept.

Derivative Rules

• Power rule: If $f(x) = x^n$, then $f^{\prime}(x) = nx^{n-1}$.
• Product rule: If $f(x) = g(x)h(x)$, then $f^{\prime}(x) = g^{\prime}(x)h(x) + g(x)h^{\prime}(x)$.
• Quotient rule: If $f(x) = \frac{g(x)}{h(x)}$, then $f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{[h(x)]^2}$.

Derivative Applications

• Physics: Derivatives are used to describe the motion of objects and to calculate forces and energies.
• Engineering: Derivatives are used to design and optimize systems, such as bridges and buildings.
• Economics: Derivatives are used to model economic systems and to make predictions about future economic trends.

Derivative Examples

• Finding the derivative of a function: $f(x) = x^2 + 3x - 4$
• Finding the derivative at a specific point: $f^{\prime}(2)$
• Using the second derivative test to determine whether a point is a maximum or minimum: $f(x) = x^3 - 6x^2 + 9x - 1$