AP Calc Notes: MD – 2 Chain Rule
Derivative of Nested Functions—see Russian nesting dolls
Review
Ex: a.
d 5
x ) = 5x4
(
dx
c.
d 5
x + sin x ) = 5x4 + cos x
(
dx
d.
d 5
x sin x ) = x5cos x + 5x4sinx
(
dx
e.
d  sin x  x5 cos x − 5x 4 sin x

=
dx  x 5 
x10
f.
d
sin ( x5 ) = ?
dx
g.
d
( sin 5 x ) = ?
dx
(
b.
d
( sin x ) = cos x
dx
)
Need a rule for the derivative of a composition of functions: y(x) = f(g(x))
f(g(x)) − f(g(a))
x→a
x−a
y'(a) = lim
f(g(x)) − f(g(a)) g(x) − g(a)
f(g(x)) − f(g(a)) g(x) − g(a)
= lim
i
i
x→a
x−a
g(x) − g(a) x→a g(x) − g(a)
x−a
= lim
f(g(x)) − f(g(a))
g(x) − g(a)
i lim
x→a
x →a
g(x) − g(a)
x−a
= lim
y'(a) = f'(g(a)) g'(a)
Since this works for any value of a, we have
d
f(g(x)) = f'(g(x))g'(x)
dx
The Chain Rule
If y = f ( u ) and u = g ( x ) , then y = f(g(x)) and
dy
d
=
f(g(x)) = f'(g(x))g'(x)
dx
dx
Derivative of
the “outside”
keep the
“inside”
Chain Rule, v 1.0
times the derivative of
the “inside”
If y = f ( u ) and u = g ( x ) , then
dy
du
and g’(x) =
so
du
dx
f’(g(x)) =
dy
dy
du
=
i
dx
du dx
Ex:
(
Chain Rule, v 2.0
)
d
sin ( x5 ) =
dx
cos(x5)(5x4) = 5x4cos(x5)
Ex:
http://brownsharpie.courtneygibbo
ns.org/?s=chain+rule
d
sin 5 x ) =
(
dx
d
(sin x)5 = 5(sin x)4cos x = 5sin4x cosx
dx
Find the derivative of the following functions:
a. f ( x ) = 4 x − 5
d
d
1
4x − 5 =
(4x - 5)1 / 2 = (4x-5) −1 / 2 (4) =
dx
dx
2
b. y = ( 3 x − 2 )
4
dy
= 4(3x - 2)3 (3) = 12(3x - 2)3
dx
c. y = tan
( x)
dy
1
sec2 x
= sec2 x
=
dx
2 x
2 x
2
4x - 5
d. y = cos 2 x
e. y =
Recall: sin 2A = 2 sin A cos A
1
16 − x 2
f. y = cos3 ( 2 x )
dy
= 3cos2 2x( − sin2x)(2) = − 6cos2 2x sin2x
dx
g. f ( x ) = tan 6 x3
π

h. f ( x ) = sin 2  3 x − 
4
