## Product Rule
General Explanation
In this lesson we will learn how to differentiate the product of two functions. First, let's derive the rule.
We'll begin with the definition of a derivative:
dy f(x+Δx) - f(x) Now, let f(x) = u(x)v(x), that is, the product of u(x) and v(x). Then, f(x+Δx) = u(x+Δx)•v(x+Δx).
dy u(x+Δx)v(x+Δx) - u(x)v(x) Separating terms gives us:
dy u(x+Δx)v(x+Δx) - u(x)v(x) We can add -u(x+Δx)v(x) + v(x)u(x+Δx), which is really 0, to the limit above:
dy u(x+Δx)v(x+Δx) - u(x+Δx)v(x) + v(x)u(x+Δx) - u(x)v(x) Let's look at the second term. If we factor out v(x) we get:
dy v(x)[u(x+Δx)-u(x)] which is equal to v(x) • du/dx. The same can be done to the first term to get u(x) • dv/dx. As a general rule:
d(u(x)•v(x)) d(v(x)) d(u(x))
Sample Problem
Find dy/dx for the following:
y = (6x
Solution
Let u(x) = 6x
^{3}+1 and v(x) = 4x^{5}. Then:
dy d(4x |