CHAPTER 9
The Chain Rule
9.1
If f(x) = x
2 + 2x — 5 and g(x) = x
3 , find formulas for the composite functions /°g and g°/.
9.7
Find the derivative of
9.8
Find the derivative of
9.9
Find the derivative of
(/°g)W=/(g(*))=/(*
3 ) = (*
3
)
2 + 2(*
3 )-5 = *
6 +2*
3 -5
(g°/)« = S(/M) = 8(*
2 + 2x-5) = (x
2 + 2x- 5)
3
9.2
Write the function
is the composition of two functions.
and let
Then
9.3
9.4
9.5
9.6
We can write
Now we can use the chain rule. Remember that
for any real number r. In particular,
By the chain rule,
We can write
By the chain rule,
Here, we have used
56
and
Use the chain rule. Think of the function as a composition (f°g)(x), where f(x) = x4 and g(x) =
Hence,
and g'(jr) = 3* -4x + 7.
Then f '(x) = 4x
3
Find the derivative of (*
3 - 2x
2 +7x- 3)
4
.
(on the right side) refers to y as a function of «.
Here, the first occurrence of y refers to y as a function of AC, while the second occurrence of y
If y = F(u) and M = G(x), then we can write y = F(G(x)). Write the chain rule formula for dyldx,
where we think of y as a function of x.
Write the chain rule formula for the derivative of f» g
(/"*)'(*) =/'(*<*))•*'(*)•
So, we must solve
Answer
and
If f(x) = 2x and g(x) = \/(x- 1), find all solutions of the equation (f°g)(x) = (g°/)(x).
4x-2 = x-l, 3x = l, *=5.
(*•/)(*) = *( A*)) = *(2*) =
(/•*)(*)= A *(*))=/
Let g(x) = 3x - 5
f(x) = Vx.
( f°*)<*) = /(gW) = A3* - 5) = V3l^5.
x
3 - 2x
2 + 7x-3.
2x
2 + 7x- 3)
3 • (3x
2 -4x + 7).
= (3*
2 + 5)4
.
x'^rr'1
»-« = - 4x -
5
.
(3x
2 + 5)"
4 = -4(3x
2 + 5K
5 • (6jr) = -
= (2x + 7)
1
'
2
.
x3 - 2x2 + 7x- 3)4 = 4(x3 -