Problem with Differentiation Using Quotient Rule

In summary, when finding the second derivative of the function h(x) = [(x^2)-1] / [2x-(x^2)], using the Quotient Rule results in the following simplified expression: h``(x) = [2x(4(x^5)-5(x^4)+(x^3)+20(x^2)-16x+4)] / [2x-(x^2)]^4. However, to check for correctness, it is recommended to graph the function and its first and second derivatives and look for key points of intersection and simplification. Additionally, a common factor of (2x-x²) should be cancelled out in the final expression.
  • #1
PianistSk8er
8
0
I am attempting to find the second derivative of a function:

h(x) = [(x^2)-1] / [2x-(x^2)]

I proceeded by using the Quotient Rule, and I found the following as the first derivative. (It is correct.)

h`(x) = [2(x^2)-2x+2] / [2x-(x^2)]^2

Next, I tried using the Quotient Rule again, and I found a very long result. I simplified the result and obtained the following:

h``(x) = [2x(4(x^5)-5(x^4)+(x^3)+20(x^2)-16x+4)] / [2x-(x^2)]^4

Is this correct? And, if so, how can I proceed from here? (The answer in the book seems to have been simplified a lot more than this one.

Thanks in advance!
Nico
 
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  • #2
Not correct, I'm afraid. You'll need to check your working.

One way of checking if your answers are correct is to graph them, using a graphing calculator or software.

h'() is the slope of h(), and h''() is the slope of h'().
If you plot h'(), you will find it has a minimum at about x = 0.8. That means its slope, the h''() curve, is zero at about x = 0.8. So if you plot h''(), and it cuts the x-axis at about x = 0.8, then that is reasonable confirmation that you may have the right answer.
Also, when doing the 2nd derivative, you should get a common factor of (2x-x²) that cancels out top and bottom. This will simplify things a bit.
 

Related to Problem with Differentiation Using Quotient Rule

What is the quotient rule in differentiation?

The quotient rule is a formula used in calculus to find the derivative of a function that is the quotient of two other functions. It states that the derivative of f(x)/g(x) is equal to (g(x)f'(x) - f(x)g'(x)) / (g(x))^2.

Why is the quotient rule important?

The quotient rule is important because it allows us to find the derivative of a function that cannot be easily simplified or broken down into simpler functions. It is also useful in solving real-world problems in fields such as physics, engineering, and economics.

What are the common mistakes when using the quotient rule?

The most common mistake when using the quotient rule is forgetting to use the minus sign in the numerator. Another mistake is not simplifying the resulting derivative after applying the rule. It is also important to remember to use the chain rule when differentiating the functions in the numerator and denominator.

Can the quotient rule be used for higher order derivatives?

Yes, the quotient rule can be extended to higher order derivatives by repeatedly applying the rule. For example, the second derivative of f(x)/g(x) is equal to (g(x)f''(x) - 2f'(x)g'(x) + f(x)g''(x)) / (g(x))^2.

Are there any other methods for differentiation besides the quotient rule?

Yes, there are several other methods for differentiation, such as the power rule, product rule, and chain rule. Each method is useful for different types of functions and can make the process of finding derivatives more efficient. It is important to understand and be familiar with all the different methods to effectively solve problems in calculus.

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