Let $y=\frac 1-2x}{3x^2}$.What Is The Value Of $\frac{dy}{dx}$ At $ X = 1 X=1 X = 1 [/tex]?Choose 1 Answer A. $-\frac{1 {3}$B. $\frac{1}{9}$C. 0D. 1

Introduction

In this article, we will explore the concept of finding the derivative of a rational function. A rational function is a function that can be expressed as the ratio of two polynomials. The derivative of a rational function is an important concept in calculus, as it allows us to study the rate of change of the function with respect to the variable.

The Given Function

The given function is:

$$y=\frac{1-2x}{3x^2}$$

We are asked to find the value of the derivative of this function at $x=1$.

Step 1: Apply the Quotient Rule

To find the derivative of the given function, we will apply the quotient rule. The quotient rule states that if we have a function of the form:

$$y=\frac{f(x)}{g(x)}$$

then the derivative of the function is given by:

$$\frac{dy}{dx}=\frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2}$$

In this case, we have:

$$f(x)=1-2x$$

and

$$g(x)=3x^2$$

Step 2: Find the Derivatives of the Numerator and Denominator

To apply the quotient rule, we need to find the derivatives of the numerator and denominator. The derivative of the numerator is:

$$f'(x)=-2$$

and the derivative of the denominator is:

$$g'(x)=6x$$

Step 3: Apply the Quotient Rule

Now that we have the derivatives of the numerator and denominator, we can apply the quotient rule:

$$\frac{dy}{dx}=\frac{(-2)(3x^2)-(1-2x)(6x)}{(3x^2)^2}$$

Simplifying the expression, we get:

$$\frac{dy}{dx}=\frac{-6x^2-6x^2+12x^2}{9x^4}$$

$$\frac{dy}{dx}=\frac{0}{9x^4}$$

$$\frac{dy}{dx}=0$$

Conclusion

In this article, we found the derivative of the given rational function using the quotient rule. We then evaluated the derivative at $x=1$ and found that the value of the derivative is 0.

Answer

The correct answer is:

C. 0

Final Answer

$$

Introduction

In our previous article, we explored the concept of finding the derivative of a rational function. A rational function is a function that can be expressed as the ratio of two polynomials. The derivative of a rational function is an important concept in calculus, as it allows us to study the rate of change of the function with respect to the variable.

Q&A Session

Q: What is a rational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials. For example, the function $y=\frac{1-2x}{3x^2}$ is a rational function.

Q: What is the quotient rule?

A: The quotient rule is a formula for finding the derivative of a rational function. It states that if we have a function of the form $y=\frac{f(x)}{g(x)}$, then the derivative of the function is given by:

$$\frac{dy}{dx}=\frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2}$$

Q: How do I apply the quotient rule?

A: To apply the quotient rule, you need to find the derivatives of the numerator and denominator. Then, you can plug these values into the formula for the quotient rule.

Q: What are the steps to find the derivative of a rational function?

A: The steps to find the derivative of a rational function are:

1. Identify the numerator and denominator of the function.
2. Find the derivatives of the numerator and denominator.
3. Apply the quotient rule using the derivatives found in step 2.
4. Simplify the expression to find the final derivative.

Q: What is the importance of finding the derivative of a rational function?

A: Finding the derivative of a rational function is important because it allows us to study the rate of change of the function with respect to the variable. This is useful in many real-world applications, such as physics, engineering, and economics.

Q: Can I use the quotient rule to find the derivative of any rational function?

A: Yes, you can use the quotient rule to find the derivative of any rational function. However, you need to make sure that the function is in the correct form, i.e., $y=\frac{f(x)}{g(x)}$.

Q: What if the denominator of the rational function is zero?

A: If the denominator of the rational function is zero, then the function is undefined at that point. In this case, you cannot find the derivative of the function using the quotient rule.

Conclusion

In this article, we answered some common questions about finding the derivative of a rational function. We hope that this Q&A session has been helpful in clarifying any doubts you may have had about this topic.

Final Answer

The final answer is $\boxed{0}$.