Let F F F Be A Function Such That F ( 1 ) = 0 F(1)=0 F ( 1 ) = 0 And F ′ ( 1 ) = − 7 F^{\prime}(1)=-7 F ′ ( 1 ) = − 7 .- Let G G G Be The Function G ( X ) = X G(x)=\sqrt{x} G ( X ) = X ​ .Evaluate $\frac{d}{d X}\left[\frac{f(x)}{g(x)}\right] At X = 1 X=1 X = 1 .

Introduction

In calculus, the quotient rule and chain rule are two fundamental concepts that help us find the derivative of composite functions. In this article, we will explore these concepts in detail and apply them to a specific problem involving two functions, $f$ and $g$. We will also evaluate the derivative of the quotient of these two functions at a given point.

The Quotient Rule

The quotient rule is a formula for finding the derivative of a quotient of two functions. It states that if we have two functions, $u(x)$ and $v(x)$, then the derivative of their quotient is given by:

$$\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{v(x)u^{\prime}(x) - u(x)v^{\prime}(x)}{[v(x)]^{2}}$$

This formula can be derived using the limit definition of a derivative.

The Chain Rule

The chain rule is a formula for finding the derivative of a composite function. It states that if we have two functions, $u(x)$ and $v(x)$, then the derivative of their composite function is given by:

$$\frac{d}{dx}[v(u(x))] = v^{\prime}(u(x)) \cdot u^{\prime}(x)$$

This formula can be derived using the limit definition of a derivative.

The Problem

Let $f$ be a function such that $f(1)=0$ and $f^{\prime}(1)=-7$. Let $g$ be the function $g(x)=\sqrt{x}$. We are asked to evaluate $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]$ at $x=1$.

Applying the Quotient Rule

To evaluate the derivative of the quotient of $f$ and $g$, we can use the quotient rule formula:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f^{\prime}(x) - f(x)g^{\prime}(x)}{[g(x)]^{2}}$$

We are given that $f(1)=0$ and $f^{\prime}(1)=-7$. We also know that $g(x)=\sqrt{x}$, so $g^{\prime}(x)=\frac{1}{2\sqrt{x}}$.

Evaluating the Derivative

Now we can substitute the values of $f(1)$, $f^{\prime}(1)$, $g(1)$, and $g^{\prime}(1)$ into the quotient rule formula:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{\sqrt{x}(-7) - 0 \cdot \frac{1}{2\sqrt{x}}}{(\sqrt{x})^{2}}$$

Simplifying the expression, we get:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{-7\sqrt{x}}{x}$$

Evaluating the Derivative at $x=1$

Finally, we can evaluate the derivative at $x=1$:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{-7\sqrt{1}}{1} = -7$$

Therefore, the derivative of the quotient of $f$ and $g$ at $x=1$ is $-7$.

Conclusion

In this article, we applied the quotient rule and chain rule to evaluate the derivative of the quotient of two functions, $f$ and $g$. We used the given values of $f(1)$, $f^{\prime}(1)$, $g(1)$, and $g^{\prime}(1)$ to simplify the expression and evaluate the derivative at $x=1$. The result is a fundamental concept in calculus that helps us understand the behavior of composite functions.

Future Directions

In future articles, we can explore more advanced topics in calculus, such as the product rule, the power rule, and the exponential rule. We can also apply these concepts to real-world problems in physics, engineering, and economics.

References

• [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
• [2] Anton, H. (2017). Calculus: A New Horizon. John Wiley & Sons.

Glossary

• Quotient Rule: A formula for finding the derivative of a quotient of two functions.
• Chain Rule: A formula for finding the derivative of a composite function.
• Derivative: A measure of how a function changes as its input changes.
• Composite Function: A function that is the result of combining two or more functions.
• Limit Definition of a Derivative: A mathematical definition of a derivative that uses limits.
  

Introduction

In our previous article, we explored the quotient rule and chain rule in calculus, and applied them to a specific problem involving two functions, $f$ and $g$. In this article, we will provide a comprehensive Q&A section to help you better understand these concepts.

Q: What is the quotient rule?

A: The quotient rule is a formula for finding the derivative of a quotient of two functions. It states that if we have two functions, $u(x)$ and $v(x)$, then the derivative of their quotient is given by:

$$\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{v(x)u^{\prime}(x) - u(x)v^{\prime}(x)}{[v(x)]^{2}}$$

Q: What is the chain rule?

A: The chain rule is a formula for finding the derivative of a composite function. It states that if we have two functions, $u(x)$ and $v(x)$, then the derivative of their composite function is given by:

$$\frac{d}{dx}[v(u(x))] = v^{\prime}(u(x)) \cdot u^{\prime}(x)$$

Q: How do I apply the quotient rule?

A: To apply the quotient rule, you need to follow these steps:

1. Identify the two functions, $u(x)$ and $v(x)$.
2. Find the derivatives of $u(x)$ and $v(x)$.
3. Substitute the values of $u(x)$, $v(x)$, $u^{\prime}(x)$, and $v^{\prime}(x)$ into the quotient rule formula.
4. Simplify the expression to find the derivative of the quotient.

Q: How do I apply the chain rule?

A: To apply the chain rule, you need to follow these steps:

1. Identify the two functions, $u(x)$ and $v(x)$.
2. Find the derivative of $v(x)$ with respect to $u(x)$.
3. Find the derivative of $u(x)$.
4. Substitute the values of $v^{\prime}(u(x))$ and $u^{\prime}(x)$ into the chain rule formula.
5. Simplify the expression to find the derivative of the composite function.

Q: What is the difference between the quotient rule and the chain rule?

A: The quotient rule is used to find the derivative of a quotient of two functions, while the chain rule is used to find the derivative of a composite function. The quotient rule involves finding the derivatives of the two functions and substituting them into a formula, while the chain rule involves finding the derivative of the outer function with respect to the inner function.

Q: Can I use the quotient rule and the chain rule together?

A: Yes, you can use the quotient rule and the chain rule together to find the derivative of a composite function that involves a quotient. For example, if you have a function like $\frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are both composite functions, you can use the quotient rule to find the derivative of the quotient, and then use the chain rule to find the derivative of the composite functions.

Q: What are some common mistakes to avoid when using the quotient rule and the chain rule?

A: Some common mistakes to avoid when using the quotient rule and the chain rule include:

• Forgetting to find the derivatives of the two functions.
• Forgetting to substitute the values of the derivatives into the formula.
• Not simplifying the expression to find the derivative.
• Using the wrong formula for the quotient rule or the chain rule.

Conclusion

In this article, we provided a comprehensive Q&A section to help you better understand the quotient rule and the chain rule in calculus. We hope that this article has been helpful in clarifying these concepts and providing you with a better understanding of how to apply them to find the derivative of composite functions.

Future Directions

In future articles, we can explore more advanced topics in calculus, such as the product rule, the power rule, and the exponential rule. We can also apply these concepts to real-world problems in physics, engineering, and economics.

References

• [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
• [2] Anton, H. (2017). Calculus: A New Horizon. John Wiley & Sons.

Glossary

• Quotient Rule: A formula for finding the derivative of a quotient of two functions.
• Chain Rule: A formula for finding the derivative of a composite function.
• Derivative: A measure of how a function changes as its input changes.
• Composite Function: A function that is the result of combining two or more functions.
• Limit Definition of a Derivative: A mathematical definition of a derivative that uses limits.