Page 6 - Calculus for the Clueless, Calc II

Let us, for completeness, recall the trig derivatives and do one longish chain rule.

Law 20

A. y= sin x, y'= cos x

B. y = cos x, y' = -sin x

C. y = tan x, y' = sec x
2
D. y = cot x, y' = -csc x
2
E. y= sec x, y'= tan x sec x

F. y = csc x, y' = -cot x csc x

Example 6—

Since this is a function of a function, we must use the extended chain rule.

6
2
5
2
Let u = tan (4x + 3x + 7). y = u and dy/du = 6u . Let v = 4x + 3x + 7. u = tan v. du/dv = sec v and dv/dx =
2
8x + 3. So
dy/dx (dy/du) times (du/dv) times (dv/dx)
=

= 6u times sec v times (8x + 3)
5
2
= [6 tan (4x + 3x + [sec (4x + 3x + 7)] (8x + 3)
2
2
2
s
7)]
Power rule—leave Derivative of trig Derivative of
trig function and function—leave crazy crazy angle
crazy angle angle untouched
untouched

You should be able to do this without substituting for u and v. It really is not that difficult with a little practice.

Law 21

A. .