AP Calc
Section 2.2: Basic Differentiation Rules and Rates
of Change
Goals for this Section:
Find the derivative of a function using the Constant Rule.
Day 1
Find the derivative of a function using the Power Rule.
Day 1
Find the derivative of a function using the Constant
Multiple Rule. Day 1
Find the derivative of a function using the Sum and
Difference Rules. Day 1
Find the derivatives of the sine function and of the
cosine function. Day 1
Use derivatives to find rates of change. Day 2
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AP Calc
Section 2.2: Basic Differentiation Rules and Rates of Change
Theorem 2.2: The Constant Rule
The derivative of a constant function is 0. That is, if c is a
real number, then
d [c]=0
dx
Example: Find the the derivative of
f(x) = 2
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AP Calc
Section 2.2: Basic Differentiation Rules and Rates of Change
Example: Find the the derivative of
f(x) = x4
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AP Calc
Section 2.2: Basic Differentiation Rules and Rates of Change
Theorem 2.3: The Power Rule
If n is a rational number, then the function f (x) = x n is
differentiable and
d [ xn ] = nxn-1
dx
For f to differentiable at x=0, n must be a number such that
xn-1 is defined on an interval containing 0.
Example: Find the the derivative of
f(x) = x7
Example: Find the the derivative of
f(x) = 1
x5
Example: Find the the derivative of
f(x) = 3 x
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AP Calc
Section 2.2: Basic Differentiation Rules and Rates of Change
Theorem 2.4: The Constant Multiple Rule
If f is a differentiable function and c is a real number then
cf is also differentiable and
d [ cf (x) ] = cf '(x)
dx
Example: Find the the derivative of
f(x) = 3x5
Example: Find the the derivative of
y=
5
(2x3)
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AP Calc
Section 2.2: Basic Differentiation Rules and Rates of Change
Theorem 2.5: The Sum & Difference Rules
The sum (or difference) of two differentiable functions f
and g is itself differentiable. Moreover, the derivative of f + g
(or f - g ) is the sum (or difference) of the derivatives of f and g
d [ f (x) + g (x) ] = f '(x) + g '(x)
dx
d [ f (x) - g (x) ] = f '(x) - g '(x)
dx
Example: Find the the derivative of
f(x) = x4 - 2x3 + 5x2 - 4x + 7
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AP Calc
Section 2.2: Basic Differentiation Rules and Rates of Change
Theorem 2.6: Derivatives of Sine & Cosine Functions
d [ sin x ] = cos x
dx
d [ cos x ] = -sin x
dx
Example: Find the the derivative of
y = x3 + 2 sin x
Example: Find the the derivative of
y = x2 - (1/2) cos x
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AP Calc
Section 2.2: Basic Differentiation Rules and Rates of Change
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AP Calc
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AP Calc
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AP Calc
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AP Calc
Section 2.2: Basic Differentiation Rules and Rates
of Change
Goals for this Section:
Find the derivative of a function using the Constant Rule.
Day 1
Find the derivative of a function using the Power Rule.
Day 1
Find the derivative of a function using the Constant
Multiple Rule. Day 1
Find the derivative of a function using the Sum and
Difference Rules. Day 1
Find the derivatives of the sine function and of the
cosine function. Day 1
Use derivatives to find rates of change. Day 2
12
AP Calc
Section 2.2: Basic Differentiation Rules and Rates of Change
Position Function: the function s that gives the position (relative
to the origin) of an object as a function of time t
Average Velocity:
Change in distance = Δ s
Change in time
Δt
(Think about D = rt)
Example: If a billiard ball is dropped from a height
of 100 feet, its height s at time t is given by the
position function
s = -16t2 + 100
where s is measured in feet and t is measured in
seconds. Find the average velocity over each of
the following time intervals:
a) [ 1 , 2 ]
b) [ 1 , 1.5 ]
c) [ 1 , 1.1 ]
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AP Calc
Section 2.2: Basic Differentiation Rules and Rates of Change
Velocity: if s = s(t) is the position function for an object
moving along a straight line, the velocity of the object at time
t is:
v (t) =
lim
∆x ➝ 0
s (t + ∆t) - s (t) = s '(t)
∆t
in other words... the velocity function is the derivative of the
position function
Speed: the absolute value of an object's velocity
Position of a "free-falling" object (neglecting air resistance):
s (t) =
2
½ gt + vot + so
where so = initial height ; v o = initial velocity ;
g = acceleration due to gravity { -32 ft/s/s or -9.8 m/s/s}
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AP Calc
Section 2.2: Basic Differentiation Rules and Rates of Change
Example: At time t = 0, a diver jumps from a
platform diving board that is 32 feet above the
water. The position of the diver is given by
s(t) = -16t2 + 16t + 32
where s is measured in feet and t is measured in
seconds.
a) When does the diver hit the water?
b) What is the diver's velocity at impact?
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