Find the derivative of e4x.
Let u = 4x. Then we can rewrite:
e4x = eu
One of the patterns says
(eu)' = eu ⋅ u'.
Since u = 4x, we know u' = 4. We substitute these into the pattern and get
(e4x)' = (eu)' = eu ⋅ u' = e4x ⋅ 4.
We can also use these patterns to find antiderivatives. The equation
(eu)' = eu ⋅ u'
means the derivative of eu is eu ⋅ u'. Thinking backwards, this means eu is an antiderivative of eu ⋅ u'. We can write the family of all antiderivatives of eu ⋅ u' as
eu + C.
Using indefinite integral notation,