Thinking Backwards - At A Glance

Hint

rearrange

Answer

• 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

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

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)

Answer

• If we let u = tan x then u' = sec² x, and we find ourselves looking at  This must have come from the natural log pattern; the original function could be ln u = ln (tan x)

Answer

• This one came from the cosine pattern. If u = 6x then u' = 6, therefore -6sin(6x) is the derivative of cos(6x).