Section 2-2: Basic Differentiation Rules and Rates of Change

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Section 2-2Basic 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

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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

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Examples of the Power Rule

• Function Derivative

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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.

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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.

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Using the Constant Multiple Rule

• Function Derivative

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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

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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

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Using the Sum and Difference Rules

• Function Derivative

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The Derivatives of Sine and Cosine Functions

• Make sure you memorize these!!!

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Using Derivatives of Sines and Cosines

• Function Derivative

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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 (cont)

• 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

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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!!