# Section 4.3 - The Chain Rule ```Math 142 Lecture Notes for Section 4.3-4.4
Section 4.3 -
1
The Chain Rule
Definition 4.3.1:
If g is a differentiable at x and f is differentiable at g(x), the the composite function h(x) = f (g(x))
is differentiable at x and it’s derivative is In Leibniz notation if y = f (u) and u = g(x) are both
differentiable functions, then
Example 4.3.2:
Use the chain rule to establish some general derivative rules:
n
If y = [f (x)] , then
If y = ef (x) , then
If y = bf (x) , then
If y = ln [f (x)], then
If y = logb [f (x)], then
Math 142 Lecture Notes for Section 4.3-4.4
Example 4.3.3:
Differentiate the following:
(a) f (x) = (3x2 + 4x)3
(b) f (x) =
(c) y =
(t2
√
2x2 − 4
−2
+ 2t + 1)3
(d) f (x) = ln(3x )
2
Math 142 Lecture Notes for Section 4.3-4.4
(e) f (x) = 8x4 ex
3
√
(f) f (x) = 3( x − ex )2
(g) f (x) = log3 (x2 + 4)
(h) f (x) = ln(
x+3
)
x−2
3
Math 142 Lecture Notes for Section 4.3-4.4
4
3
3 ln x2 − x4 )
(i) f (x) =
x2
Example 4.3.4:
Find the value(s) of x where the tangent line is horizontal for
x3
(5x − 4)2
Example 4.3.5:
Given the following information g(2) = 6, g 0 (2) = 3, f 0 (6) = 5, f 0 (3) = 1, f 0 (2) = 7. If h(x) =
f (g(x)), then what is h0 (2)?
Math 142 Lecture Notes for Section 4.3-4.4
5
Example 4.3.6:
Let y = ln u and u = 3x2 +
5
dy
. Use the second form of the derivative to find
.
x
dx
Example 4.3.7:
If James invests 10,000 into a money market account offered by Skyfall Ltd. with an annual interest
rate of 7.5% compounded continuously, how fast is the balance growing after 25 years?
Suggested Homework: (4.3) 1-37(odd), 41,45. (4.4) 1-73(odd).
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