# A question about understanding f',f''

• May 31st 2009, 05:32 PM
jimmyp
So I've been given the function:

$\displaystyle f(x)=3x^3-9x$ and I found $\displaystyle f' , f''$ 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 $\displaystyle f$ and its successive derivatives?
• May 31st 2009, 06:35 PM
VonNemo19
Quote:

Originally Posted by jimmyp
So I've been given the function:

$\displaystyle f(x)=3x^3-9x$ and I found $\displaystyle f' , f''$ 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 $\displaystyle f$ 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 $\displaystyle \frac{dy}{dx}[f(x)+g(x)]=\frac{dy}{dx}f(x)+\frac{dy}{dx}g(x)$. 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: $\displaystyle \frac{dy}{dx}ax^n=n*ax^{n-1}$

Therefore, a polynomial of the form $\displaystyle a_0x^n+a_1x^{n-1}+...+a_{n-1}x+a_n$ when diferrenyiated becomes $\displaystyle na_0^{n-1}+(n-1)a_{1}^{n-2}+...+a_{n-1}+0$, 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