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Math Help - A question about understanding f',f''

  1. #1
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    A question about understanding f',f''

    So I've been given the function:

    f(x)=3x^3-9x and I found f' , f'' and I graphed them all together but I'm trying to understand: in general what is the relationship among the degrees of a polynomial function and the degress of its successive derivatives?


    Also what is the releshonship among the degrees of f and its successive derivatives?
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  2. #2
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    Quote Originally Posted by jimmyp View Post
    So I've been given the function:

    f(x)=3x^3-9x and I found f' , f'' and I graphed them all together but I'm trying to understand: in general what is the relationship among the degrees of a polynomial function and the degress of its successive derivatives?


    Also what is the releshonship among the degrees of f and its successive derivatives?

    Well, by the addition property of differentiation we know that the derivative of a sum is the sum of the derviatives \frac{dy}{dx}[f(x)+g(x)]=\frac{dy}{dx}f(x)+\frac{dy}{dx}g(x). So, when we differentiate a polynomial we know that all we do is differentiate each term of the polynomial and take the sum. Furthermore, the power rule states that when differentiating a term in a polynomial, all we have to do is the following: \frac{dy}{dx}ax^n=n*ax^{n-1}

    Therefore, a polynomial of the form a_0x^n+a_1x^{n-1}+...+a_{n-1}x+a_n when diferrenyiated becomes na_0^{n-1}+(n-1)a_{1}^{n-2}+...+a_{n-1}+0, and it is seen that in each successive derivative, the leading coeficient is reduced by 1.

    A prrof of the power rule and addition property can be found here:

    Calculus with polynomials - Wikipedia, the free encyclopedia
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