section 2-2: basic differentiation rules and rates of change eun jin choi, victoria jaques, mark...
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Section 2-2:Basic Differentiation Rules
and Rates of Change
Eun Jin Choi,Victoria Jaques,Mark Anthony Russ
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Brief Overview
The Constant Rule Power Rule Constant Multiple Rule Sum and Difference Rules Derivatives of Sine and Cosine Functions How to find Rates of Change (Velocity and
Acceleration)
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The Constant Rule
The derivative of a constant function is 0. That is, if c is a real number, then
0cdxd
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Examples of the Constant Rule
Function Derivative y = 34 dy/dx = 0 y = 2 y’ = 0 s(t)= -3 s’(t) = 0
Notice the different notations for derivatives. You get the idea!!!
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The Power Rule
If n is a rational number, then the function f(x) = xn is differentiable and
)(' 1 nnxxf
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Examples of the Power Rule
Function Derivative
23)(' xxf
3)( xxg 32
32
31
3
131
)('x
xxdxd
xg
2
1)(
xxh 3
32 22
xxx
dxd
dxdh
3)( xxf
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Finding the Slope at a Point
In order to do this, you must first take the derivative of the equation.
Then, plug in the point that is given at x. Example:
Find the slope of the graph of x4 at -1.
4)1(4)1('
4)(')(3
34
f
xxfxxf
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The Constant Multiple Rule
If f is a differentiable function and c is a real number, then cf is also differentiable and
So, pretty much for this rule, if the function has a constant in front of the variable, you just have to factor it out and then differentiate the function.
)()( xfdxdcxfc
dxd
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Using the Constant Multiple Rule
Function Derivative
x2 2
211 21222
xxx
dxd
xdxd
dxdy
54 2t ttt
dtd
tdtd
tf 58
542
542
54 )2()('
3 22
1
x
35
35
32
3
132
21
21
xxx
dxd
dxdy
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Using Parentheses when Differentiating
This is the same as the Constant Multiple Rule, but it can look a lot more organized!
Examples:Original Rewrite Differentiate Simplify
325x
325 xy 4
25 3' xy 42
15'
xy
2)3(7
x263xy )2(63' xy xy 126'
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The Sum and Difference Rules
The sum (or difference) of two differentiable functions is differentiable.
The derivative of the sum of two functions is the sum of their derivatives.
Sum (Difference) Rule:
)(')(')()( xgxfxgxfdxd
)(')(')()( xgxfxgxfdxd
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Using the Sum and Difference Rules
Function Derivative
54)( 3 xxxf 43)(' 2 xxf
xxx
xg 232
)( 34
292)(' 23 xxxg
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The Derivatives of Sine and Cosine Functions
Make sure you memorize these!!!
xxdxd cossin
xxdxd sincos
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Using Derivatives of Sines and Cosines
Function Derivative
xsin2 xcos2
2sin x
2cos x
xx cos xsin1
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Rates of Change
Applications involving rates of change include population growth rates, production rates, water flow rates, velocity, and acceleration.
Velocity = distance / time Average Velocity = ∆distance / ∆time
Acceleration = velocity / time Average Acceleration = ∆velocity / ∆time
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Rates of Change (con’t)
In a nutshell, when you are given a function expressing the position (distance) of an object, to find the velocity you must take the derivative of the position function and then plug in the point you are trying to find.
Likewise, if you are trying to find the acceleration, you must take the derivative of the velocity function and then plug in the point you are trying to find.
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Using the Derivative to Find Velocity
Usual position function:
– s0 = initial position
– v0 = initial velocity
– g = acceleration due to gravity (-32 ft/sec2 or -9.8 m/sec2)
Example: Find the velocity at 2 seconds of an object with position s(t) = -16t2 + 20t + 32.
– First take the derivative: s’(t) = -32t + 20– Then, plug in 2 to find the answer: s’(2) = -44 ft/sec
002
21)( stvgtts
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Congratulations!!!
You have now mastered Section 2 of Chapter 2 in your very fine Calculus Book: Calculus of a Single Variable 7th Edition!!