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3.3
Rules for Differentiation
Quick Review
12 .1 12 xx1
2 1 .2
x
x
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In Exercises 1 – 6, write the expression as a power of x.
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Quick Review06252equation theof roots positive theFind .7 23 xxx
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ax
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ax
lim .
Quick Review. ofrespect with functions theseof derivative theFind .9 x
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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
Essential QuestionWhat are the rules to help us find derivatives of functionsanalytically 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
11xf
0 xf
5xxf
45xxf
Rule 3: The Constant Multiple Rule
Rule 4: The Sum and Difference Rule.
23xxf xxf 23
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
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.
42 25 xxxf
45xxf x4
Example Rules for Differentiation
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242 if Find .1 23 bbba
db
da
232 bdb
da b24
3
2
26b b83
2
Example Rules for Differentiation
tangents?horizontalany have 392
1 curve theDoes .2 34 xxy
342
1xy 239 x
32x 227x0
where?so, If
0 2x 272 x
02 x 0272 x
0x2
27x
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
2 4 if Find .3 23 xxxfxf
43 xy x2 22 x 23x42x x8 43x 26x
45x 26x x8
5x 32x 24x 8
Rule 6: The Quotient Rule
y.graphicallSupport .1
ateDifferenti .43
2
x
xxf
1 3xy
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.
x2 2x 23x
231 x
2x
42x 43x
231 x
23
4
1
2
x
xx
Example Rules for Differentiation
.2 ,1point theandorigin at the 1
4 ,Serpentine sNewton' to tangents theFind .5
2
x
xy
12
xy
4 2x 4 x4 x2
22 1x
4 28x
22 1x 22
2
1
44
x
x
At x = 0. 21
4m 4
0y 4 0x
xy 4
At x = 1. 22
0m 0
2y 0 1x2y
Rule 7: Power Rule for Negative Integer Powers of x
2). (1,point at the 2
3 curve the toline tangent for theequation theFind .6
2
x
xy
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
2
xy 1
2
3 x
2
1y 2
2
3 x
22
3
2
1
x
At x = 1
2
3
2
1m 1
2y 1 1x
3 xy2y 1 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.52 of sderivativefour first theFind .7 234 xxxxy
34xy 23x x4 1212xy x6 4
xy 24 624y
Pg. 92, 2.4 #2-40 even
