2.4 (Part
I)
The
Product Rule
As you saw in your last lab, the product rule is:
.
If we let u = f(x) and v = g(x), then the product rule can be written more compactly as:
,
or if we let let u’ = du and v’ = dv, this can be written as:
.
Think of du as a very small change in u, and dv as a very small change in v. Then this last formula shows how the product uv changes if u and v change both change a little. The following diagram illustrates this.
In this diagram, the blue region has area uv, and the change in uv is the outer border, including the red, yellow, and black regions. So geometrically, the change in uv is:
.
What about that dudv term, which isn't part of the product rule?? Remember, du and dv are VERY small (infinitesimals). When you multiply two very small numbers, you get a REALLY tiny number. For example, if du and dv were on the order of one millionth, then dudv is on the order of one millionth of one millionth. Thus dudv is so small, it essentially vanishes!
If this seems too much like hand-waving, please refer to the actual proof! J
Example 1:
We could multiply out those trinomials, but the product rule allows us to compute the derivative directly:
One could of course simplify this further.
Example 2: Suppose that
and let
We could have asked that last question, without even introducing the function h, by asking:
That vertical line with the x=a at the bottom means to evaluate the expression at x=a.
Click here to go to part II of this section, Higher-Order Derivatives.
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