Derivative of fraction without quotient rule

In summary, the problem asks to find the derivative of the function \frac{x^3-3x^2+4}{x^2}, without using the quotient or product rule. After manipulating the function and taking its derivative, the result obtained was \frac{-7}{x^3}. However, the correct answer given in the book is \frac{x^3-8}{x^3}. It was later realized that there was a mistake in the algebra, but the calculus was done correctly.
  • #1
mark1
27
0
The following problem appears in my textbook (before it discusses the quotient or product rule, so those rules cannot be used for the answer):

Find the derivative of the function: [tex]\frac{x^3-3x^2+4}{x^2}[/tex]

I brought the denominator to the top and multiplied it out to get [tex]{x-3+4x^-2}[/ltex]. I then took the derivative of that to get [tex]{1-0-8x^-3}[/tex], which can be simplified to [tex]\frac{-7}{x^3}[/tex].

However, in the back of my book, the answer is given as [tex]\frac{x^3-8}{x^3}[/tex].

Please enlighten me as to where i went wrong.
 
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  • #2
Your arithmetic:
[tex]1 - \frac{8}{x^3} \neq \frac{-7}{x^3}[/tex]
 
  • #3
Thanks! For some reason, i did the calculus right but messed up on the algebra. :redface:
 
  • #4
It happens. :smile:
 

1. What is the derivative of a fraction without using the quotient rule?

The derivative of a fraction can be found by using the product rule, where the numerator and denominator are treated as separate functions and then multiplied together. This can also be written as the quotient of the derivative of the numerator and the derivative of the denominator.

2. Why would I want to find the derivative of a fraction without using the quotient rule?

The quotient rule can be more complex and time consuming to use compared to the product rule. By finding the derivative of a fraction without using the quotient rule, it can save time and make the problem simpler to solve.

3. Can I always use the product rule to find the derivative of a fraction?

No, the product rule can only be used when the fraction is written as a product of two functions. If the fraction is written as a quotient, then the quotient rule must be used instead.

4. What is the difference between the quotient rule and the product rule?

The quotient rule is used to find the derivative of a quotient, where the numerator and denominator are treated as separate functions. The product rule is used to find the derivative of a product, where two functions are multiplied together.

5. Are there any other methods for finding the derivative of a fraction?

Yes, there are other methods such as the chain rule and the power rule which can also be used to find the derivative of a fraction. However, depending on the specific problem, one method may be more efficient or easier to use than others.

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