3.3 rules for differentiation. what you’ll learn about positive integer powers, multiples, sums...
TRANSCRIPT
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