Derivatives of Polynomials

Suggested Prerequisites: Definition of differentiation,

Polynomials are some of the simplest functions we use. We need to know ther derivatives of polynomials such as x4+3x, 8x2+3x+6, and 2. Let's start with the easiest of these, the function y=f(x)=c, where c is any constant, such as 2, 15.4, or one million and four. It turns out that the derivative of any constant function is zero. This makes sense if you think about the derivative as the slope of a tangent line.

What about the derivative of the sum of functions? Happily, the derivative of a sum is the sum of the derivative.

The derivative of a constant multiple times a function is the constant times the derivative of the function:

We also need to know the derivative of xn for any value of n. For positive integer values of n, we can use the binomial theorem:

In the final step, all term go to zero except the first, leaving us with

Now we're ready to take the derivative of any polynomial function that's thrown at us.

Examples

  4. And there's no reason that the variables x is special:

Exercises:

    Find the derivative with respect to x of the following functions:

Solutions to the exercises | Back to the Calculus page | Back to the World Web Math top page
jjnichol@mit.edu
Last modified 23 June 1997