Derivative of Monomials

Our Introductory Calculus tutorial software program contains over 40 topic areas. One of them is Derivative of Monomials, and this page summarizes the main ideas of this topic. This page is intended for review, and is not a substitute for the interactive, self-paced tutorials of the MathTutor introductory calculus software program.

General Explanation

In this lesson we will learn how to find the derivative of a term such as axn. Let's derive the rule now.
By definition:


dy        f(x+Δx) - f(x)
= lim
dx Δx→0 Δx


Now, let's let f(x) = axn. Then, f(x+Δx) = a(x+Δx)n.


dy        a(x+Δx)n - axn
= lim
dx Δx→0 Δx


Using the binomial theorem, we can expand a(x+Δx)n into axn + anxn-1Δx ... + aΔxn.


dy        axn + anxn-1Δx + ... + aΔxn - axn
= lim
dx Δx→0 Δx


Notice the axn and -axn terms cancel. We can factor out the Δx terms from the numerator and cancel it with the Δx term in the denominator.


= lim anxn-1 + ... + aΔxn-1
dx Δx→0


As Δx approaches zero, all the terms containing Δx will approach zero. The limit of the above expression is anxn-1.
The general rule for the derivative of a monomial can be written as:


= anxn-1


Sample Problem







Using the general rule for the derivative of a monomial:


dy   d(9x6)
= = 9 * 6 * x6-1 = 54x5
dx dx