## Derivative of Monomials
General Explanation
In this lesson we will learn how to find the derivative of a term such as ax
^{n}. Let's derive the rule now.
By definition:
dy f(x+Δx) - f(x) Now, let's let f(x) = ax ^{n}. Then, f(x+Δx) = a(x+Δx)^{n}.
dy a(x+Δx) Using the binomial theorem, we can expand a(x+Δx) ^{n} into ax^{n} + anx^{n-1}Δx ... + aΔx^{n}.
dy ax Notice the ax ^{n} and -ax^{n} terms cancel. We can factor out the Δx terms from the numerator and cancel it with the Δx term in the denominator.
dy As Δx approaches zero, all the terms containing Δx will approach zero. The limit of the above expression is anx ^{n-1}.
The general rule for the derivative of a monomial can be written as:
d(ax
Sample Problem
Find:
d(9x
Solution
Using the general rule for the derivative of a monomial:
dy d(9x |