If $g(x)=\ln \left(x^3-7 X^2\right$\], Find $g^{\prime}(x$\].A. $g^{\prime}(x)=\frac{x^3-7 X^2}{3 X^2-14 X}$B. $g^{\prime}(x)=\frac{1}{3 X^2-14 X}$C. $g^{\prime}(x)=\frac{3 X^2-14 X}{x^3-7 X^2}$D.

Introduction

In calculus, the derivative of a function is a measure of how the function changes as its input changes. In this article, we will explore the derivative of a natural logarithmic function, specifically the function $g(x)=\ln \left(x^3-7 x^2\right)$. We will use the chain rule and the quotient rule to find the derivative of this function.

The Chain Rule

The chain rule is a fundamental rule in calculus that allows us to find the derivative of a composite function. A composite function is a function that is composed of two or more functions. The chain rule states that if we have a composite function $f(g(x))$, then the derivative of this function is given by:

$$\frac{d}{dx}f(g(x))=f^{\prime}(g(x))\cdot g^{\prime}(x)$$

The Quotient Rule

The quotient rule is another fundamental rule in calculus that allows us to find the derivative of a quotient of two functions. The quotient rule states that if we have a quotient of two functions $f(x)/g(x)$, then the derivative of this function is given by:

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

Finding the Derivative of $g(x)$

To find the derivative of $g(x)=\ln \left(x^3-7 x^2\right)$, we will use the chain rule and the quotient rule. First, we will find the derivative of the inner function $x^3-7 x^2$. Using the power rule, we get:

$$\frac{d}{dx}(x^3-7 x^2)=3x^2-14x$$

Next, we will find the derivative of the outer function $\ln(x)$. Using the chain rule, we get:

$$\frac{d}{dx}\ln(x)=\frac{1}{x}$$

Now, we can use the quotient rule to find the derivative of $g(x)$. We have:

$$g^{\prime}(x)=\frac{1}{x^3-7 x^2}\cdot (3x^2-14x)$$

Simplifying this expression, we get:

$$g^{\prime}(x)=\frac{3x^2-14x}{x^3-7 x^2}$$

Conclusion

In this article, we have found the derivative of the natural logarithmic function $g(x)=\ln \left(x^3-7 x^2\right)$. We used the chain rule and the quotient rule to find the derivative of this function. The derivative of $g(x)$ is given by:

$$g^{\prime}(x)=\frac{3x^2-14x}{x^3-7 x^2}$$

This result is consistent with option C.

Discussion

The derivative of a natural logarithmic function is an important concept in calculus. It is used to find the rate of change of a function and to solve optimization problems. In this article, we have used the chain rule and the quotient rule to find the derivative of a natural logarithmic function. We have also discussed the importance of the derivative in calculus.

Final Answer

The final answer is:

g^{\prime}(x)=\frac{3x^2-14x}{x^3-7 x^2}$<br/> **Derivatives of Natural Logarithmic Functions: Q&A** ===================================================== **Introduction** --------------- In our previous article, we explored the derivative of a natural logarithmic function, specifically the function $g(x)=\ln \left(x^3-7 x^2\right)$. We used the chain rule and the quotient rule to find the derivative of this function. In this article, we will answer some common questions related to the derivative of natural logarithmic functions. **Q: What is the derivative of a natural logarithmic function?** --------------------------------------------------------- A: The derivative of a natural logarithmic function is given by: $\frac{d}{dx}\ln(x)=\frac{1}{x}

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

A: To find the derivative of a natural logarithmic function, you can use the chain rule and the quotient rule. First, find the derivative of the inner function using the power rule. Then, find the derivative of the outer function using the chain rule. Finally, use the quotient rule to find the derivative of the natural logarithmic function.

Q: What is the chain rule?

A: The chain rule is a fundamental rule in calculus that allows us to find the derivative of a composite function. A composite function is a function that is composed of two or more functions. The chain rule states that if we have a composite function $f(g(x))$, then the derivative of this function is given by:

$$\frac{d}{dx}f(g(x))=f^{\prime}(g(x))\cdot g^{\prime}(x)$$

Q: What is the quotient rule?

A: The quotient rule is another fundamental rule in calculus that allows us to find the derivative of a quotient of two functions. The quotient rule states that if we have a quotient of two functions $f(x)/g(x)$, then the derivative of this function is given by:

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

Q: How do I apply the chain rule and the quotient rule to find the derivative of a natural logarithmic function?

A: To apply the chain rule and the quotient rule to find the derivative of a natural logarithmic function, follow these steps:

1. Find the derivative of the inner function using the power rule.
2. Find the derivative of the outer function using the chain rule.
3. Use the quotient rule to find the derivative of the natural logarithmic function.

Q: What are some common mistakes to avoid when finding the derivative of a natural logarithmic function?

A: Some common mistakes to avoid when finding the derivative of a natural logarithmic function include:

• Forgetting to apply the chain rule and the quotient rule.
• Not simplifying the expression correctly.
• Not checking the domain of the function.

Conclusion

In this article, we have answered some common questions related to the derivative of natural logarithmic functions. We have discussed the chain rule and the quotient rule, and provided examples of how to apply these rules to find the derivative of a natural logarithmic function. We have also discussed some common mistakes to avoid when finding the derivative of a natural logarithmic function.

Final Answer

The final answer is:

$$g^{\prime}(x)=\frac{3x^2-14x}{x^3-7 x^2}$$