Universiti Pendidikan Sultan Idris

1. Beginning Calculus
- Chain Rule and Higher Derivatives -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,
FSMT - UPSI
(LECTURE SLIDES SERIES)
VillaRINO DoMath, FSMT-UPSI
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2. The Chain Rule Higher Derivatives
Learning Outcomes
Apply chain rule to compute the derivatives.
Find the higher derivatives.
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3. The Chain Rule Higher Derivatives
The Chain Rule
De…nition 1
If g is di¤erentiable at x and f is di¤erentiable at g (x) , then f g is
di¤erentiable at x and
(f g)0
(x) = f 0
[g (x)] g0
(x) (1)
(f g) (x) = f [g (x)] is considered as the "outside function" and
g (x) is the "inside function".
The derivative of (f g) (x) is the derivative of the outside function
multiply the derivative of the inside function.
In Leibniz notation, if y = f (u) and u = g (x) are both
di¤erentiable functions, then
dy
dx
=
dy
du
du
dx
(2)
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4. The Chain Rule Higher Derivatives
Example
Let h (x) = x3 1
100
. Note that:
h (x) = (f g) (x) = f [g (x)]
where f (x) = x100 is the outside function and g (x) = x3 1 is the
inside function. So,
h0
(x) = (f g)0
(x)
= f 0
[g (x)] g0
(x)
= 100 x3
1
99
3x2
= 300x2
x3
1
99
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5. The Chain Rule Higher Derivatives
Example - continue
Using Leibniz notation:
y = x3 1
100
. Let u = x3 1. Then y = u100. So,
dy
dx
=
dy
du
du
dx
= 100u99
3x2
= 300x2
x3
1
99
VillaRINO DoMath, FSMT-UPSI
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6. The Chain Rule Higher Derivatives
Example
d
dx
(sec x) = tan x sec x
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7. The Chain Rule Higher Derivatives
Example
y =
p
sec x3.
Let u = x3; v = sec u. Then, y =
p
v.
dy
dx
=
dy
dv
dv
du
du
dx
or
d
dx
p
sec x3 =
1
2
p
v
(sec u tan u) 3x2
=
3x2
2
p
sec x3
sec x3
tan x3
VillaRINO DoMath, FSMT-UPSI
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8. The Chain Rule Higher Derivatives
Using Chan Rule and Product Rule to Di¤erentiate The
Quotient Rule
u
v
0
=
u0v v0u
v2
u
v
0
= uv 1
0
= u0
v 1
+ v 1
0
u
= u0
v 1
v 2
v0
u
=
u0v v0u
v2
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9. The Chain Rule Higher Derivatives
Higher Derivatives
Let y = f (x)
The derivative of f :
y0
= f 0
(x) =
dy
dx
=
df
dx
=
d
dx
(y) =
d
dx
f (x)
The second derivative of f :
y00
= f 00
(x) =
d2y
dx2
=
df 0
dx
=
d
dx
dy
dx
=
d
dx
f 0
(x)
The nth derivative of f :
y(n)
= f (n)
(x) =
d(n)y
dx(n)
=
df (n)
dx
=
d
dx
d(n 1)y
dx(n 1)
!
=
d
dx
f (n 1)
(x)
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10. The Chain Rule Higher Derivatives
Example
d(n)
dx(n)
(xn) = n (n 1) (n 2) 2 1 = n!
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11. The Chain Rule Higher Derivatives
Example
d2
dx2
p
x + 2 =
1
4 3
p
x + 2
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