Finding the Derivative of x(3-x^2)^-2

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In summary, the problem is to find the derivative of x(3-x^2)^-2 using the chain rule. However, the solution also requires the use of the product rule or quotient rule due to the presence of two terms involving x being multiplied together. The correct answer is 4x^2 *(3-x^2)^-3 + (3-x^2)^-2. The mistake made was only finding one term instead of both.
  • #1

Homework Statement

I am supposed to find the derivative of: x(3-x^2)^-2

Homework Equations

The chain rule

The Attempt at a Solution

Well I feel that I am good at using the chain rule but something tells me I can't use it here, because when I do, I only get about half of the answer.

But anyway, I multiplied x by -2 , which I multiplied by the group (3-x^2)^-3. Then I multiplied that term by the derivative of the first group, (3-x^2), and got: 4x^2 * (3-x^-2)^-3

however, the right answer is listed as: 4x^2 *(3-x^2)^-3 + (3-x^2)^-2

for some reason I don't think the chain rule applies to this problem? or perhaps I am doing it wrong... I would appreciate any help or explanation
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  • #2
Use the product rule ;-) (or quotient rule, if you prefer) It generates two terms, but you only found one of them.
  • #3
Like diazona says, the product rule (combined with your chain rule) shall set you free! :smile:

If it was simply


then the chain rule would have sufficed.

However, you have two terms involving x that are multiplied with each other so you also need to incorporate the product rule (or quotient rule for this particular case, but I'd personally prefer the product rule).
  • #4
Thank you very much guys! I see where I went wrong. I appreciate your help, thanks again :)

1. What is the derivative of x(3-x^2)^-2?

The derivative of x(3-x^2)^-2 is -2x/(3-x^2)^3.

2. How do you find the derivative of x(3-x^2)^-2?

To find the derivative of x(3-x^2)^-2, you can use the power rule and the chain rule. First, rewrite the function as x(3-x^2)^-2 = x(3-x^2)^-1 * (3-x^2)^-1. Then, use the power rule to find the derivative of (3-x^2)^-1, which is -2x/(3-x^2)^2. Finally, use the chain rule to find the derivative of (3-x^2)^-1, which is -2x/(3-x^2)^3.

3. Can you simplify the derivative of x(3-x^2)^-2?

Yes, the derivative of x(3-x^2)^-2 can be simplified to -2x/(3-x^2)^3. You can also expand the function and simplify further to get -2x/(9-6x^2+x^4).

4. How does the derivative of x(3-x^2)^-2 relate to the original function?

The derivative of x(3-x^2)^-2 is the slope of the tangent line to the original function at any given point. It represents the rate of change of the function at that point.

5. Can you provide an example of using the derivative of x(3-x^2)^-2 in real life?

The derivative of x(3-x^2)^-2 can be used in physics to calculate the velocity of a moving object. If x represents time and (3-x^2)^-2 represents the position of the object, then the derivative -2x/(3-x^2)^3 represents the velocity of the object at any given time. This can be helpful in understanding the motion of objects in real life scenarios.

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