Find The Derivative Of The Function:$\[ Y = \frac{x^2 - 3x + 2}{x^7 - 2} \\]A) $\[ Y^{\prime} = \frac{-5x^8 + 19x^7 - 14x^6 - 4x + 6}{(x^7 - 2)^2} \\]B) $\[ Y^{\prime} = \frac{-5x^8 + 18x^7 - 14x^6 - 3x + 6}{(x^7 - 2)^2}

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Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. In this article, we will focus on finding the derivative of a rational function, which is a function that can be expressed as the ratio of two polynomials. We will use the quotient rule to find the derivative of the given function.

The Quotient Rule

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

where f(x) and g(x) are both functions of x, then the derivative of y with respect to x is given by:

y′=f′(x)g(x)−f(x)g′(x)(g(x))2{ y^{\prime} = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2} }

The Given Function

The given function is:

y=x2−3x+2x7−2{ y = \frac{x^2 - 3x + 2}{x^7 - 2} }

To find the derivative of this function, we will use the quotient rule.

Finding the Derivative

Using the quotient rule, we have:

y′=f′(x)g(x)−f(x)g′(x)(g(x))2{ y^{\prime} = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2} }

where f(x) = x^2 - 3x + 2 and g(x) = x^7 - 2.

First, we need to find the derivatives of f(x) and g(x):

f′(x)=2x−3{ f^{\prime}(x) = 2x - 3 }

g′(x)=7x6{ g^{\prime}(x) = 7x^6 }

Now, we can plug these values into the quotient rule formula:

y′=(2x−3)(x7−2)−(x2−3x+2)(7x6)(x7−2)2{ y^{\prime} = \frac{(2x - 3)(x^7 - 2) - (x^2 - 3x + 2)(7x^6)}{(x^7 - 2)^2} }

To simplify this expression, we can expand the numerator and denominator:

y′=2x8−6x7+4x−6−7x8+21x6−14x6+21x5−14x4+6x3−12x2+12(x7−2)2{ y^{\prime} = \frac{2x^8 - 6x^7 + 4x - 6 - 7x^8 + 21x^6 - 14x^6 + 21x^5 - 14x^4 + 6x^3 - 12x^2 + 12}{(x^7 - 2)^2} }

Combining like terms, we get:

y′=−5x8+19x7−14x6−4x+6(x7−2)2{ y^{\prime} = \frac{-5x^8 + 19x^7 - 14x^6 - 4x + 6}{(x^7 - 2)^2} }

Conclusion

In this article, we used the quotient rule to find the derivative of a rational function. We started with the given function and applied the quotient rule formula to find the derivative. We then simplified the expression to get the final answer.

Answer

The derivative of the given function is:

y′=−5x8+19x7−14x6−4x+6(x7−2)2{ y^{\prime} = \frac{-5x^8 + 19x^7 - 14x^6 - 4x + 6}{(x^7 - 2)^2} }

This is the correct answer, which matches option A.

Discussion

The quotient rule is a powerful tool for finding the derivative of rational functions. It allows us to find the derivative of a function that cannot be easily differentiated using other rules. In this article, we used the quotient rule to find the derivative of a rational function, and we got the correct answer.

Example Problems

Here are some example problems that you can try to practice finding the derivative of rational functions using the quotient rule:

  1. Find the derivative of the function:

y=x3+2x2−3x−1x4+2x3−3x2−2x+1{ y = \frac{x^3 + 2x^2 - 3x - 1}{x^4 + 2x^3 - 3x^2 - 2x + 1} }

  1. Find the derivative of the function:

y=x2−4x+3x3−2x2+x−1{ y = \frac{x^2 - 4x + 3}{x^3 - 2x^2 + x - 1} }

  1. Find the derivative of the function:

y=x4+2x3−3x2−x+1x5+2x4−3x3−2x2+x+1{ y = \frac{x^4 + 2x^3 - 3x^2 - x + 1}{x^5 + 2x^4 - 3x^3 - 2x^2 + x + 1} }

Try to solve these problems using the quotient rule, and see if you get the correct answers.

References

  • Calculus by Michael Spivak
  • Calculus by James Stewart
  • Calculus by David Guichard

Introduction

In our previous article, we discussed how to find the derivative of a rational function using the quotient rule. In this article, we will answer some common questions that students often have when it comes to finding the derivative of rational functions.

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

where f(x) and g(x) are both functions of x, then the derivative of y with respect to x is given by:

y′=f′(x)g(x)−f(x)g′(x)(g(x))2{ y^{\prime} = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{(g(x))^2} }

Q: How do I apply the quotient rule?

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

  1. Identify the numerator and denominator of the rational function.
  2. Find the derivatives of the numerator and denominator.
  3. Plug the derivatives into the quotient rule formula.
  4. Simplify the expression to get the final answer.

Q: What if the numerator and denominator have a common factor?

A: If the numerator and denominator have a common factor, you can cancel it out before applying the quotient rule. This will simplify the expression and make it easier to work with.

Q: Can I use the quotient rule to find the derivative of a rational function with a negative exponent?

A: Yes, you can use the quotient rule to find the derivative of a rational function with a negative exponent. However, you need to be careful when simplifying the expression, as the negative exponent can affect the final answer.

Q: How do I know when to use the quotient rule versus the product rule or chain rule?

A: The quotient rule is used when you have a rational function, which is a function that can be expressed as the ratio of two polynomials. The product rule is used when you have a product of two functions, and the chain rule is used when you have a composite function. You need to identify the type of function you are working with and choose the appropriate rule to apply.

Q: Can I use the quotient rule to find the derivative of a rational function with a variable in the denominator?

A: Yes, you can use the quotient rule to find the derivative of a rational function with a variable in the denominator. However, you need to be careful when simplifying the expression, as the variable in the denominator can affect the final answer.

Q: How do I check my work when using the quotient rule?

A: To check your work when using the quotient rule, you can plug the original function and its derivative into the quotient rule formula and simplify the expression. If the final answer matches the original function, then you have done the problem correctly.

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

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

  • Forgetting to cancel out common factors between the numerator and denominator.
  • Not simplifying the expression correctly.
  • Not checking the final answer to make sure it matches the original function.

Conclusion

In this article, we answered some common questions that students often have when it comes to finding the derivative of rational functions using the quotient rule. We hope that this article has been helpful in clarifying any confusion you may have had about the quotient rule and its application.