MATH 131-505 Spring 2015
3.4
c
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3.4 The Chain Rule
The Chain Rule: If g is differentiable at x and f is differentiable at g(x), then the composite
function F = f ◦ g defined by F (x) = f (g(x)) is differentiable at x and F 0 is given by the product
F 0 (x) = f 0 (g(x)) · g 0 (x)
In Leibniz notation, if y = f (u) and u = g(x) are both differentiable functions, then
dy
dy du
=
dx
du dx
Example 1: (p. 205) If F (x) = f (g(x)), where f (−2) = 8, f 0 (−2) = 4, f 0 (5) = 3, g(5) = −2, and
g 0 (5) = 6, find F 0 (5).
The Power Rule Combined with the Chain Rule: If n is any real number and u = g(x) is
differentiable, then
d n
du
(u ) = nun−1
dx
dx
Alternatively,
d
(g(x))n = n (g(x))n−1 g 0 (x)
dx
Examples: Find the derivative of the function.
1. f (x) = (4x − ax2 )100
2. h(x) = ex cos x
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MATH 131-505 Spring 2015
r
3. g(x) =
3
t2
3.4
c
Wen
Liu
t
+4
4. y = (2x − 5)4 (8x2 − 5)−3
5. y = cos(a3 + x3 )
Note:
d x
(a ) = ax ln a
dx
Example 6: Find f 0 if f (x) = 43c sin(πx) .
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