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Rule 1: Derivative of a Constant The first rule of differentiation is that the derivative of every constant function is the zero of the function. If f(x) = c, then Proof:
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AP CalculusChapter 3 Section 3RULES FOR DIFFERENTIATION
Rule 1: Derivative of a Constant The first rule of differentiation is that the derivative of every constant function is the zero of the function.
If f(x) = c, then
Proof:
0 cdxd
dxdf
00limlim)()(lim000
hhh hcc
hxfhxf
Rule 2: Power Rule for Positive Integer Powers of x If n is a positive integer, then
The proof is on page 116.
1)( nn nxxdxd
Rule 2: Examples
34
23
12
4
3
2
xxdxd
xxdxd
xxdxd
Rule 3: Constant Multiple Rule If u is a differentiable function of x and c is a constant, then
Example:
dxduccu
dxd
)(
3344 284777 xxxdxdx
dxd
Rule 4: Sum & Difference Rule If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points,
Example:
This is a very powerful rule because it allows us to find the derivatives of polynomials.
dxdv
dxduvu
dxd
xxdxdx
dxdx
dxdxx
dxd 442222 32424
Rule 5: Product Rule The product of 2 differentiable functions u and v is differentiable, and
“the first times the derivative of the second plus the second times the derivative of the first”
Note: the order you do the differentiation does not matter because we are adding the two together.
dxduv
dxdvuuv
dxd
Product Rule Example Find the derivative of Solution: Let
Rule 6: Quotient Rule At a point where v ≠ 0, the quotient y = u/v of 2 differentiable functions is differentiable, and
“the bottom times the derivative of the top minus the top times the derivative of the bottom, all over the bottom squared”
2vdxdvu
dxduv
vu
dxd
Quotient Rule Example Differentiate Solution: Let
Rule 7: Power Rule for Negative Integer Powers of x If n is a negative integer and x ≠ 0, then
We will use this rule after we discuss the Chain Rule. This rule can be used to solve for a derivative in place of the Quotient Rule because we can redefine the denominator as a negative integer power of x.
1 nn nxxdxd
Second & Higher Order Derivatives We can find the derivative of the derivative, which is called the second derivative. Denoted:
We can also find the third, fourth, fifth, and nth derivatives. Denoted:
2
2
dxyd
dxdy
dxd
dxydy
n
nnn
dxydy
dxdy
dxyd
dxydy
)1()(
3
3
Example Find the first four derivatives of y = x3 – 5x2 + 2
Example Find the first four derivatives of y = x3 – 5x2 + 2
0
6106103
)4(
2
y
yxy
xxy
Finding Instantaneous Rate of Change
An orange farmer currently has 200 trees yielding an average of 15 bushels of oranges per tree. She is expanding her farm at the rate of 15 trees per year, while improved husbandry is improving her average annual yield by 1.2 bushels per tree. What is the current (instantaneous) rate of increase of her total annual production of oranges?
Solution Let the functions t and y be defined as follows:
Then, We know: We need to find Apply Product Rule:
Homework # 3 – 52 by multiples of 3 on pages 124 - 125
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