1) The document discusses the chain rule and higher derivatives in calculus. It defines the chain rule and provides examples of applying it to find the derivative of composite functions. 2) It also explains how to take higher derivatives by applying the derivative operator multiple times, and gives an example of finding the nth derivative of xn. 3) Additional examples are provided of using the chain rule to find derivatives of more complex expressions involving radicals, quotients, and other functions.

1. Beginning Calculus
- Chain Rule and Higher Derivatives -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,
FSMT - UPSI
(LECTURE SLIDES SERIES)
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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
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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
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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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