Rearrange a little: -3sin x(cos x)2 = 3(cos x)2 · (-sin x). If u = cos x, then this looks much like 3u2 · u' which came from the power function pattern. A function with this derivative is u3, or (cos x)3
Example 2
Determine a function whose derivative is
6(x3 + 2x + 1)5(3x2 + 2)
Answer
This one also came from the power function pattern. Let u = x3 + 2x + 1. Then u' = 3x2 + 2, therefore
6(x3 + 2x + 1)5(3x2 + 2)
is the same thing as
6u5 · u'.
A function with this derivative is u6, or
(x3 + 2x + 1)6
Example 3
Determine a function whose derivative is
cos(x + 4)
Answer
The only reasonable thing to do is let u= x + 4. Then u' = 1, therefore
cos(x + 4) = cos(x + 4) · 1 = cos u · u'.
A function with this derivative is sin u = sin(x + 4)
Example 4
Determine a function whose derivative is
Answer
If we let u = tan x then u' = sec2x, and we find ourselves looking at This must have come from the natural log pattern; the original function could be
lnu = ln (tan x)
Example 5
Determine a function whose derivative is
-6sin(6x)
Answer
This one came from the cosine pattern. If u = 6x then u' = 6, therefore -6sin(6x)
is the derivative of
cos(6x).
This must have come from the natural log pattern; the original function could be