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Calculus 3.3 - University of Houston

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3.3 Rules for Differentiation
Colorado National Monument
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington
If the derivative of a function is its slope, then for a
constant function, the derivative must be zero.
d
dx
пЂЁcпЂ© пЂЅ 0
example:
yпЂЅ3
yп‚ў пЂЅ 0
The derivative of a constant is zero.
п‚®
If we find derivatives with the difference quotient:
d
x пЂЅ lim
2
пЂЁx пЂ« hпЂ©
hп‚® 0
dx
2
пЂ­x
пЂЁ x пЂ« 2 xh пЂ« h
2
2
пЂЅ lim
hп‚® 0
h
2
пЂ©пЂ­ x
2
пЂЅ 2x
h
1
1
d
x пЂЅ lim
3
пЂЁx пЂ« hпЂ©
hп‚® 0
dx
пЂЅ lim
пЂЁx
3
3
пЂ­x
h
пЂ« 3 x h пЂ« 3 xh пЂ« h
2
2
2
3
1
пЂ©пЂ­ x
3
1
пЂЅ 3x
2
2
3
3
4
пЂЁ x пЂ« 4 x h пЂ« 6 x h пЂ« 4 xh пЂ« h
x пЂЅ lim
4
3
hп‚® 0
We observe a pattern:
2
2
3
4
5
2
1
3
6
1
4
10 10
1
5
1
(Pascal’s Triangle)
h
4
dx
1
1
hп‚® 0
d
3
1
пЂ©пЂ­ x
4
пЂЅ 4x
3
h
2 x 3x
2
4x
3
5x
4
6x
5
…
п‚®
We observe a pattern:
d
dx
пЂЁx
n
пЂ©
2 x 3x
2
4x
3
5x
4
6x
5
…
examples:
пЂЅ nx
n пЂ­1
f
пЂЁxпЂ© пЂЅ
x
yпЂЅ x
4
f п‚ўпЂЁ xпЂ© пЂЅ 4x
3
8
yп‚ў пЂЅ 8 x
7
power rule
п‚®
constant multiple rule:
d
dx
пЂЁ cu пЂ© пЂЅ c
du
dx
examples:
d
cx пЂЅ cnx
n
n пЂ­1
dx
d
7 x пЂЅ 7 пѓ— 5 x пЂЅ 35 x
5
4
4
dx
When we used the difference quotient, we observed that
since the limit had no effect on a constant coefficient, that
the constant could be factored to the outside.
п‚®
constant multiple rule:
d
dx
пЂЁ cu пЂ© пЂЅ c
du
dx
sum and difference rules:
d
dx
пЂЁu пЂ« v пЂ© пЂЅ
du
пЂ«
dx
y пЂЅ x пЂ« 12 x
4
y п‚ў пЂЅ 4 x пЂ« 12
3
dv
d
dx
dx
пЂЁu пЂ­ v пЂ© пЂЅ
du
dx
пЂ­
dv
dx
y пЂЅ x пЂ­ 2x пЂ« 2
4
2
(Each term
dyis treated separately)
пЂЅ 4x пЂ­ 4x
3
dx
п‚®
Example:
Find the horizontal tangents of:
y пЂЅ x пЂ­ 2x пЂ« 2
4
dy
2
пЂЅ 4x пЂ­ 4x
3
dx
Horizontal tangents occur when slope = zero.
4x пЂ­ 4x пЂЅ 0
3
x пЂ­xпЂЅ0
3
x пЂЁ x пЂ­ 1пЂ© пЂЅ 0
2
x пЂЁ x пЂ« 1пЂ© пЂЁ x пЂ­ 1пЂ© пЂЅ 0
Plugging the x values into the
original equation, we get:
y пЂЅ 2, y пЂЅ 1, y пЂЅ 1
(The function is even, so we
only get two horizontal
tangents.)
x пЂЅ 0, пЂ­ 1, 1
п‚®
4
3
2
1
-2
-1
0
-1
-2
1
2
4
y пЂЅ x пЂ­ 2x пЂ« 2
4
3
2
1
-2
-1
0
-1
-2
1
2
2
4
y пЂЅ x пЂ­ 2x пЂ« 2
4
3
yпЂЅ2
2
1
-2
-1
0
-1
-2
1
2
2
4
y пЂЅ x пЂ­ 2x пЂ« 2
4
3
yпЂЅ2
2
y пЂЅ1
1
-2
-1
0
-1
-2
1
2
2
4
y пЂЅ x пЂ­ 2x пЂ« 2
4
3
2
1
-2
-1
0
-1
-2
1
2
2
4
y пЂЅ x пЂ­ 2x пЂ« 2
4
2
3
2
dy
1
пЂЅ 4x пЂ­ 4x
3
dx
-2
-1
0
1
2
-1
First derivative
(slope) is zero at:
-2
x пЂЅ 0, пЂ­ 1, 1
п‚®
product rule:
d
dx
пЂЁ uv пЂ© пЂЅ u
dv
пЂ«v
dx
du
Notice that this is not just the
product of two derivatives.
dx
This is sometimes memorized as: d пЂЁ u v пЂ© пЂЅ u d v пЂ« v d u
d
пЂЁ
пѓ© x2 пЂ« 3
dx пѓ«
d
пЂЁ
dx
пЂ©пЂЁ
пЂ© пЂЁ
3
2x пЂ« 5x пѓ№ пЂЅ x2 пЂ« 3
пѓ»
2 x пЂ« 5 x пЂ« 6 x пЂ« 15 x
5
d
2x
пЂЁ
dx
3
5
3
пЂ« 11 x пЂ« 15 x
3
10 x пЂ« 33 x пЂ« 15
4
2
пЂ©
пЂ©
пЂЁ
пЂЁ6x пЂ« 5пЂ©пЂ« 2x пЂ« 5x
2
3
пЂ©пЂЁ 2 x пЂ©
пЂ©
6 x пЂ« 5 x пЂ« 18 x пЂ« 15 пЂ« 4 x пЂ« 10 x
4
2
2
4
2
10 x пЂ« 33 x пЂ« 15
4
2
п‚®
quotient rule:
d пѓ¦uпѓ¶
пѓ§ пѓ·пЂЅ
dx пѓЁ v пѓё
v
du
d 2x пЂ« 5x
пЂ­u
dx
dx
v
dx
x пЂ«3
пЂЅ
пѓ¦ u пѓ¶ v du пЂ­ u dv
dпѓ§ пѓ·пЂЅ
2
v
v
пѓЁ пѓё
or
2
пЂЁ x пЂ« 3пЂ© пЂЁ6 x пЂ« 5пЂ© пЂ­ пЂЁ 2 x пЂ« 5x пЂ© пЂЁ 2 x пЂ©
2
3
2
dv
2
3
пЂЁx
2
пЂ« 3пЂ©
2
п‚®
Higher Order Derivatives:
yп‚ў пЂЅ
dy
is the first derivative of y with respect to x.
dx
y п‚ўп‚ў пЂЅ
dy п‚ў
пЂЅ
dx
y п‚ўп‚ўп‚ў пЂЅ
d dy
dx dx
dy п‚ўп‚ў
2
пЂЅ
d y
dx
2
is the second derivative.
(y double prime)
is the third derivative.
dx
y
пЂЁ4пЂ©
пЂЅ
d
dx
y п‚ўп‚ўп‚ў
is the fourth derivative.
We will learn
later what these
higher order
derivatives are
used for.
p
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