How to Differentiate Polynomials

Polynomials are mathematical expressions consisting of a finite sum of variables with the form a_i*(x^i) where a_i is a nonzero constant, x is the variable and i is an integer greater than or equal to zero. Polynomials are used in many different branches of mathematics, including differential calculus. The following steps will show how to differentiate polynomials.

Instructions

Difficulty: Challenging

1. Express a polynomial as x_n*(x^n) + x_n-1*(x^(n-1)) + … + x_i*(x^i) + ... + x_0. Determine the derivative of each term x_i*(x^i) and then add these terms together to find the derivative of the entire polynomial.
2. Define the derivative of a function f(x) as the limit of (f(x+h) – f(x))/h as h approaches zero. The process of finding the derivative of a function is known as differentiation.
3. Substitute x_i*(x^i) for f(x) in Step 1. We now have the derivative of x_i*(x^i) = the limit of (x_i*(x+h)^i – x_i*x^i)/h as h approaches zero. This term is equal to x_i*((x+h)^i – x^i)/h.
4. Use the binomial theorem to show that (x+h)^i = (x^i) + (i)(x^(i-1))h + … + h^i and therefore the derivative of x_i*(x^i) = x_i * the limit of (x^i) + (i)(x^(i-1))h + … + h^i – x^i)/h as h approaches zero. Note that all but the first two terms in this equation will contain a factor of h^j where j is greater than or equal to 2.
5. Cancel the terms x^i and –x^i in the last equation given in Step 3 to yield (i)(x^(i-1))h + … + h^i). Divide these terms through by h. Because all but the first term has at least one factor h, the limit of all but the first term approaches zero as h approaches zero.
6. Evaluate the derivative of x_i*(x^i) = (x_i)(i)(x^(i-1)). We can now evaluate the derivative of the polynomial x_n*(x^n) + x_n-1*(x^(n-1)) + … + x_i*(x^i) + ... + x_0 as x_n(n)(x^(n-1)) + x_n-1(n-1)(x^(n-2) + … + x_i*(i)(x^(i-1)) + … + x_1.