Finding the nth derivative of a function

In summary, the conversation discusses finding a formula for the nth derivative of the function f(x)=x1/3. The formula involves alternating signs and an exponent of x(1/3-n), but the coefficient of x is difficult to determine. The conversation suggests working with the general form f(x)=x^p, then substituting p=1/3 at the end for easier calculation. Keeping everything in terms of p instead of numbers can prevent confusion and make it easier to look at more terms.
  • #1

Homework Statement


I'm trying to find a formula for the nth derivative for the function f(x)=x1/3

The Attempt at a Solution


I know that it has alternating signs so it start with (-1)n+1 and I know the exponent for it is x(1/3-n) but I'm having a hard time figuring out the coefficient of x.

For the fourth derivative I have 1/3(1/3-1)(1/3-2)(1/3-3)x(1/3-4)The example our teacher gave us was x-1 which was much easier in my opinion...
 
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  • #2
Try expressing the powers and the coefficients in a fractional form while taking derivatives and see if the pattern becomes clearer that way.
 
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  • #3
I rewrote it three different ways and I'm still having a hard time seeing the complete pattern.

I'm using the fourth derivative: f4(x)=(1/3)(-2/3)(-5/3)(-8/3)x-11/3
and: f4(x)=(1/3)(1/3-1)(1/3-2)(1/3-3)x-11/3
and:f4(x)=(1/3)(1/3-3/3)(1/3-6/3)(1/3-9/3)x-11/3

So far I have fn(x)=(-1)n+1(?/3n)x1/3-n
 
  • #4
TheRainMan713 said:

Homework Statement


I'm trying to find a formula for the nth derivative for the function f(x)=x1/3

The Attempt at a Solution


I know that it has alternating signs so it start with (-1)n+1 and I know the exponent for it is x(1/3-n) but I'm having a hard time figuring out the coefficient of x.

For the fourth derivative I have 1/3(1/3-1)(1/3-2)(1/3-3)x(1/3-4)The example our teacher gave us was x-1 which was much easier in my opinion...

IMHO it is much easier to work with the general form ##f(x) = x^p##, then take ##p = 1/3## after all the work is finished.

The reason it is easier is that you have symbolic factors such as ##p - 1##, ##p-2,## etc. and by keeping them symbolic you can keep straight the different "effects". For example, if you see a number like ##-2## somewhere in your calculation, it is not easy to know if it really is a ##``-2"## or a ##``-1 - 3/3"## or ##``-6/3"##---and sometimes that matters a lot when you want to look at more terms, etc. By keeping everything in terms of ##p## there is never any chance of confusion.
 

1. What is the nth derivative of a function?

The nth derivative of a function is the derivative of the derivative of the function, n times. It represents the rate of change of a function after n times of differentiation.

2. How is the nth derivative of a function calculated?

The nth derivative of a function is calculated by taking the derivative of the function n times using the appropriate differentiation rules. For example, if the function is f(x), the nth derivative would be denoted as f(n)(x).

3. Why is finding the nth derivative of a function important?

Finding the nth derivative of a function is important in various fields of science and engineering, such as physics, economics, and statistics. It allows us to analyze the behavior and characteristics of a function at different levels of complexity and can provide valuable insights into the behavior of systems.

4. What is the difference between the nth derivative and the n+1th derivative of a function?

The nth derivative and the n+1th derivative of a function are different in that the nth derivative represents the rate of change after n times of differentiation, while the n+1th derivative represents the rate of change after n+1 times of differentiation. In other words, the n+1th derivative is one level of complexity higher than the nth derivative.

5. Can the nth derivative of a function be negative?

Yes, the nth derivative of a function can be negative, positive, or zero depending on the function and the value of n. A negative nth derivative indicates that the function is decreasing at a certain point, while a positive nth derivative indicates that the function is increasing at that point.

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