The Chain Rule

1
The Chain Rule
is called the Power Rule, and recall that I said
cant be done by the power rule because the base
is an expression more complicated than x. In
other words, in order to use the power rule, the
base must be x, or the variable you are
differentiating with respect to. However, it
doesnt mean we cant differentiate (2x 1)3.
All we need is a rule called the Chain Rule, more
appropriately, Chain Rule with the Power Rule
(three versions are provided, its up to you to
choose the one you like) 1. 2. 3.
Here is how it works 1. 2. 3. If f(x) 3(6
5x2)4, find f ?(x).
Product Rule, Quotient Rule and Chain Rule (lets
throw them together) Find f ?(x) for each of
the following functions 1. f(x) (x 2)2(x
3)3 2. 3.
2
Derivatives of Trigonometric FunctionsDerivative
of sin x
If f(x) sin x, what is f ?(x)? Recall the
limit definition of derivative
Chain Rule on sin (g(x)) If f(x) sin (g(x)),
then f ?(x) cos (g(x))?g?(x). That is, d/dxsin
(expression) cos (expression)?d/dx(expression).
Examples For each of the following functions,
find its derivative. 1. f(x) sin x2 ? f ?(x)
2. f(t) sin (t 2)(3t2 4) ? 3. g(x)
?
3
Derivatives of Trigonometric FunctionsDerivative
of cos x
If f(x) cos x, what is f ?(x)? This time we are
not going to use the limit definition to find f
?(x), but rather, recall cos x sin ( )
Chain Rule on cos (g(x)) If f(x) cos (g(x)),
then f ?(x) ____________. That is, d/dxcos
(expression) _________________________. Exampl
es For each of the following functions, find its
derivative. 1. f(x) cos (x2 2x 1) ? f ?(x)
2. f(t) cos (t 3)(2t2 1) ? 3. g(x)
?
4
Derivatives of Trigonometric FunctionsDerivatives
of the Other Four
If f(x) tan x, what is f ?(x)? Recall
If f(x) cot x, what is f ?(x)? Recall
If f(x) sec x, what is f ?(x)? Recall
If f(x) csc x, what is f ?(x)? Recall
Chain Rule on these functions If f(x) tan
(g(x)), then f ?(x) ______________________ If
f(x) cot (g(x)), then f ?(x)
______________________ If f(x) sec (g(x)),
then f ?(x) ______________________ If f(x)
csc (g(x)), then f ?(x) ______________________
Examples For each of the following functions,
find its derivative. 1. f(x) tan x2 sin 2x ? f
?(x) 2. f(t) csc (t 3) cot (2t2 1) ? 3.
g(x) ?
5
Summary of Derivatives of All Six and Precaution
on Chain Rules
The table on the right shows the derivatives of
the six basic trig. functions (notice the
derivatives of the three cofunctionscosine,
cotangent and cosecanthave a ________ sign). Of
course, Chain Rule can be applied to each one of
them (see bottom table). The order of applying
Chain Rules We have to apply Chain Rule for
finding the derivative of many functions, and for
some of them, we need to apply Chain Rule
more than once, and the ORDER we apply the Chain
Rule MATTERS. Examples Find the derivatives of
the following functions.
f(x) sin x3 vs. f(x) sin 3 x f(x)
tan 2 (cos x) vs. f(x) tan (cos 2 x) vs.
f(x) tan (cos x2)
f(x) cot 2 sin(x2 3)