1
Chain Rule for the Slope of f .g.x//
Chain Rule for the Slope of f .g.x//
y D g.x/ z D f .y/
y Dx
5
Average slope
��
��
the chain is z D f .g.x//
the chain is z D .x 5 /4 D x 20
z
z
y
D
Just cancel y
x
y
x
zDy
4
dz
dz dy
D
D CHAIN RULE (like cancelling dy)
dx dy dx
You MUST change y to g.x/ in the final answer
Instant slope
Example of chain
z D y 4 D .x 5 /4
dz
D 4y 3
dy
dy
D 5x 4
dx
dz
dz
dy
D
D .4y 3 /.5x 4 / D 20y 3 x 4
dx
dy
dx
dz
D 20.x 5 /3 x 4 D 20 x 19
Replace y by x 5 to get only x
dx
dz
CHECK z D .x 5 /4 D x 20 does have
D 20 x 19
dx
dz
dz
1. Find
for z D cos .4x/
Write y D 4x and z D cos y so
D
dx
dx
Chain rule
2. Find
CHECK
dz
for z D .1 C 4x/2
dx
Write y D 1 C 4x and z D y 2 so
.1 C 4x/2 D 1 C 8x C 16x 2 so
dz
D
dx
Practice Questions
3. Find
dh
for h.x/ D .sin 3x/.cos 3x/
dx
Product rule first
Then the Chain rule for each factor
dh
d
d
D .sin 3x/ .cos 3x/ C .cos 3x/ .sin 3x/
dx
dx
dx
D .sin 3x/.CHAIN/ C .cos 3x/.CHAIN/ D ‹
dz
D
dx
2
Chain Rule for the Slope of f .g.x//
4. Tough challenge: Find the second derivative of z.x/ D f .g.x//
dz
dz
dy
FIRST
Function of y.x/
D
DERIV
times function of x
dx
dy
dx
d 2z
dz d dy
dy d dz
PRODUCT
D
C
RULE
dx 2
dy dx dx
dx dx dy
2 2 dy
d z
dy
dy
dz
d y
SECOND
C
twiceŠ
DERIV
dy
dx 2
dx
dy 2
dx
dx
Check y D x 5
SECOND
DERIV
z D y 4 D x 20
dz
D 20x 19
dx
d 2z
D 380x 18
dx 2
.4y 3 /.20x 3 / C .5x 4 /.12y 2 /.5x 4 /
80 C 300 D 380 OK
MIT OpenCourseWare
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Resource: Highlights of Calculus
Gilbert Strang
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