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3.3
Rules for Differentiation
What you’ll learn about Positive Integer Powers, Multiples,
Sums and Differences Products and Quotients Negative Integer Powers of x Second and Higher Order Derivatives
… and whyThese rules help us find derivatives of functions analytically
in a more efficient way.
Rule 1 Derivative of a Constant Function
If is the function with the constant value , then
0
This means that the derivative of every constant function
is the zero function.
f c
df dc
dx dx
Rule 2 Power Rule for Positive Integer Powers of x.
1
If is a positive integer, then
The Power Rule says:
To differentiate , multiply by and subtract 1 from the exponent.
n n
n
n
dx nx
dx
x n
Rule 3 The Constant Multiple Rule
If is a differentiable function of and is a constant, then
This says that if a differentiable function is multiplied by a constant,
then its derivative is multiplied by the same cons
u x c
d ducu c
dx dx
tant.
Rule 4 The Sum and Difference Rule
If and are differentiable functions of , then their sum and differences
are differentiable at every point where and are differentiable. At such points,
.
u v x
u v
d du dvu v
dx dx dx
Example Positive Integer Powers, Multiples, Sums, and Differences 4 2 3
Differentiate the polynomial 2 194
That is, find .
y x x x
dy
dx
4 2Sum and Difference Rule
3Constant and Power Rules
3
By Rule 4 we can differentiate the polynomial term-by-term,
applying Rules 1 through 3.
32 19
4
34 2 2 0
43
= 4 44
dy d d d dx x x
dx dx dx dx dx
x x
x x
Example Positive Integer Powers, Multiples, Sums, and Differences
4 2Does the curve 8 2 have any horizontal tangents?
If so, where do they occur?
Verify you result by graphing the function.
y x x
4 2 3
If any horizontal tangents exist, they will occur where the slope
is equal to zero. To find these points we will set 0 and solve for .
Calculate 8 2 4 16
Set 0 and solve
dy
dxdy
xdx
dy dx x x x
dx dxdy
dx
3
2 2
for
4 16 0
4 4 0; 4 0 4 0
This gives horizontal tangents at 0, 2, 2.
x
x x
x x x x
x
Rule 5 The Product Rule
The product of two differentiable functions and is differentiable, and
The derivative of a product is actually the sum of two products.
u v
d dv duuv u v
dx dx dx
Example Using the Product Rule 3 2Find if 4 3f x f x x x
3 2
3 2 3 2 2
4 4 2
4 2
Using the Product Rule with 4 and 3,gives
4 3 4 2 3 3
2 8 3 9
5 9 8
u x v x
df x x x x x x x
dx
x x x x
x x x
Rule 6 The Quotient Rule
2
At a point where 0, the quotient of two differentiable
functions is differentiable, and
Since order is important in subtraction, be sure to set up the
numerator of the
uv y
v
du dvv ud u dx dx
dx v v
Quotient rule correctly.
Example Using the Quotient Rule
3
2
4Find if
3
xf x f x
x
3 2
3 2 2 3
22 2
4 2 4
22
4 2
22
Using the Quotient Rule with 4 and 3,gives
4 3 3 4 2
3 3
3 9 2 8
3
9 8
3
u x v x
x x x x xdf x
dx x x
x x x x
x
x x x
x
Rule 7 Power Rule for Negative Integer Powers of x
1
If is a negative integer and 0, then
.
This is basically the same as Rule 2 except now is negative.
n n
n x
dx nx
dxn
Example Negative Integer Powers of x
1Find an equation for the line tangent to the curve at the point 1,1 .y
x
1
22
Rewrite the function as and use the Power Rule to
find the derivative.
11
1Evaluate 1 = 1
1The line through 1,1 with slope 1 is
1 1 1
2
This shows the graph of the funct
y x
y xx
y
m
y x
y x
ion and its tangent line at (1, 1).
1yx
2y x
Second and Higher Order Derivatives
2
2
The derivative is called the of with respect to .
The first derivative may itself be a differentiable function of . If so,
its derivative, ,
dyy first derivative y x
dxx
dy d dy d yy
dx dx dx dx
3
3
is called the of with respect to . If
double prime is differentiable, its derivative,
,
is called the of with respect to .
second derivative y x y
y
dy d yy
dx dxthird derivative y x
Second and Higher Order Derivatives
1
The multiple-prime notation begins to lose its usefulness after three primes.
So we use " super "
to denote the th derivative of with respect to .
Do not confuse the notation with th
n n
n
dy y y n
dxn y x
y
e th power of , which is . nn y y
Quick Quiz Sections 3.1 – 3.3
You may use a graphing calculator to solve the following problems.
1. Let 1 . Which of the following statements about are true?
I. is continuous at 1.
II. is differentiable at 1.
III. has
f x x f
f x
f x
f
a corner at 1.
A I only
B II only
C III only
D I and III only
x
Quick Quiz Sections 3.1 – 3.3
You may use a graphing calculator to solve the following problems.
1. Let 1 . Which of the following statements about are true?
I. is continuous at 1.
II. is differentiable at 1.
III. has
f x x f
f x
f x
f
a corner at 1.
A I only
B II only
D
C III only
I and III only
x
Quick Quiz Sections 3.1 – 3.3
2. If the line normal to the graph of at the point 1,2 passes through
the point 1,1 , then which of the following gives the value of 1 ?
A 2
B 2
1C
21
D 2
E 3
f
f
Quick Quiz Sections 3.1 – 3.3
2. If the line normal to the graph of at the point 1,2 passes through
the point 1,1 , then which of the following gives the value of 1 ?
A 2
B 2
1C
12
E
D
32
f
f
Quick Quiz Sections 3.1 – 3.3
2
2
2
2
4 33. Find if .
2 110
A4 3
10B
4 3
10C
2 1
10D
2 1
E 2
dy xy
dx x
x
x
x
x
Quick Quiz Sections 3.1 – 3.3
2
2
2
2
1
4 33. Find if .
2 1
10B
4 3
10C
2 1
10D
2 1
E
0A
4
2
3x
dy xy
dx x
x
x
x