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Chapter 3: Differentiation

Section 3.1: The Derivative as a Function

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. Section 2.6
    2. solve polynomial equations
    3. solve rational equations
    4. tangent line to a curve
    5. point-slope equation of a line
  2. procedural knowledge
    1. how to evaluate the limit of a function
    2. how to determine if a function is continuous at a point
  3. conditional knowledge
    1. know when to use the appropriate methods for evaluating limits

Learning Goals

  1. declarative knowledge (definitions)
    1. derivative of a function at x
    2. differentiable
    3. differentiation
    4. right-hand derivative of a function at a
    5. left-hand derivative of a function at b
    6. the various notations for the derivative
  2. procedural knowledge
    1. find the derivative of a function at x
    2. find the equation of a tangent line to a curve at a point
    3. find when a function has a horizontal tangent line
    4. find when a function has a vertical tangent line
  3. conditional knowledge
    1. know why differentiability implies continuity
    2. know continuity does not guarantee differentiability
    3. identify where a function is not differentiable
    4. know how to identify the graph of a derivative of a function with the graph of the function
 

Section 3.2: Differentiation Rules for Polynomials, Exponentials, Products, and Quotients

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. solve polynomial equations
    2. solve rational equations
    3. tangent line to a curve
    4. perpendicular lines
    5. point-slope equation of a line
    6. limit laws (to understand the proofs only)
    7. derivative as a function
  2. procedural knowledge
    1. how to use the limit definition to find the derivative of a function
  3. conditional knowledge
    1. none

Learning Goals

  1. declarative knowledge (definitions)
    1. derivative of a constant function
    2. Power Rule for differentiation
    3. Constant Multiple Rule for differentiation
    4. Derivative Sum Rule
    5. Derivative of the Natural Exponential Function, i.e., ex
    6. Derivative Product Rule
    7. Derivative Quotient Rule
    8. second derivative
    9. higher order derivatives and their notations
    10. normal to a curve
  2. procedural knowledge
    1. how to use the various rules to find the derivative of a function
    2. how to use the various rules to find the second derivative of a function
    3. find the equation of the line tangent to a function at a point
    4. find the equation of the line normal to a function at a point
    5. find where a function has horizontal tangent lines
    6. find where a function has tangent lines with a specified slope
  3. conditional knowledge
    1. know the difference between a constant and a coefficient
    2. determine the most efficient methid for finding the derivative
    3. find the derivative of a function with variables other than x and x
 

Section 3.3: The Derivative as a Rate of Change

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. Section 2.1
    2. slope of a secant line
    3. slope of a tangent line
    4. point-slope equation of a line
  2. procedural knowledge
    1. how to find the slope between two points
    2. how to find slope of the line tangent to a curve
  3. conditional knowledge
    1. know the difference between slope of a secant line and slope of a tangent line

Learning Goals

  1. declarative knowledge (definitions)
    1. instantaneous rate of change (i.e., derivative)
    2. average rate of change
    3. displacement, s
    4. total distance, |s|
    5. average velocity, vavg
    6. [instantaneous] velocity, v = s′
    7. speed, |v| = |s′|
    8. acceleration, a = v′ = s′′
    9. jerk, j = a′ = v′′ = s′′′
    10. average cost (or revenue or profit), a.k.a., the difference quotient of the cost (or revenue or profit) function
    11. marginal cost (or revenue or profit), a.k.a., the derivative of the cost (or revenue or profit) function
  2. procedural knowledge
    1. given s(t), find displacement on [a,b]
    2. given s(t), find average velocity on [a,b]
    3. given s(t), find speed at endpoints of [a,b]
    4. given s(t), find acceleration at endpoints of [a,b]
    5. given s(t), find when body changes direction on [a,b], i.e., when v(t) changes sign
    6. given v(t), find average velocity on [a,b]
    7. given v(t), find speed at endpoints of [a,b]
    8. given v(t), find acceleration at time t
    9. given v(t), find when body changes direction on [a,b]
    10. given v(t), find when body is moving forward/backward on [a,b]
    11. given v(t), find when body's velocity is increasing/decreasing on [a,b]
    12. given a(t), find when body is speeding up/slowing down on [a,b]
    13. given a(t), find when body's acceleration is increasing/decreasing on [a,b]
    14. given s(t) for an object moving vertically with/against gravity, find when body reaches its maximum height (i.e., set v(t) = 0 and solve for t) and the value of the maxmimum height (i.e., evaluate s(t))
    15. find any of the above from the graphs of s(t), v(t), or a(t)
    16. identify s(t), v(t), a(t) from the graphs of the three functions
    17. find average cost (or revenue or profit) from the cost (or revenue or profit) function
    18. find marginal cost (or revenue or profit) from the cost (or revenue or profit) function
    19. solve other rate of change questions
  3. conditional knowledge
    1. identify what function is given and what is being asked
    2. know how to identify the graph of a derivative of a function with the graph of the function
 

Section 3.4: Derivatives of Trigonometric Functions

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. definitions of trigonometric functions on the right triangle
    2. definitions of trigonometric functions on the unit circle
    3. trigonometric identities
  2. procedural knowledge
    1. how to simplify/rearrange trigonometric expressions
    2. evaluate the trigonometric ratio for any multiple of reference angles
  3. conditional knowledge
    1. none

Learning Goals

  1. declarative knowledge (definitions)
    1. differentiation rules for trigonometric functions
  2. procedural knowledge
    1. find the derivative of a function containing trigonometric expressions
    2. find the equation of the line tangent to a function at a point
    3. find where a function has horizontal tangent lines
    4. find where a function has tangent lines with a specified slope
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives
 

Section 3.5: The Chain Rule and Parametric Equations

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. composition of functions
  2. procedural knowledge
    1. how to find the component functions of a composition of functions
    2. how to simplify/rearrange trigonometric expressions
    3. evaluate the trigonometric ratio for any multiple of reference angles
  3. conditional knowledge
    1. determine the 'order' of the component functions in a composition of functions

Learning Goals

  1. declarative knowledge (definitions)
    1. Power Chain Rule
    2. parametric curve
    3. parametric equations
    4. parameter for the parametric curve
    5. parameter interval
    6. initial point
    7. terminal point
    8. parametrization of the curve
    9. Parametric Formula for dy/dx
    10. Parametric Formula for d2y/dx2
  2. procedural knowledge
    1. find the derivative of composition of functions
    2. find the multiple derivatives of composition of functions
    3. find the derivative of parametric equations
    4. rewrite parametric equations into Cartesian equation
    5. rewrite Cartesian equation in parametric form
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives
    2. identify the difference between product of functions and a composition of functions
    3. determine the 'order' of the component functions in a composition of functions
 

Section 3.6: Implicit Differentiation

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. differentiation rules from previous sections
  2. procedural knowledge
    1. evaluate the trigonometric ratio for any multiple of reference angles
    2. find the derivative of a function using any previous rule(s)
    3. rewrite parametric equations into Cartesian equation
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives

Learning Goals

  1. declarative knowledge (definitions)
    1. implicitly defined functions
    2. implicit differentiation
  2. procedural knowledge
    1. find the derivative of any order of an implicitly defined function
    2. find the equation of the line tangent to an implicitly defined function at a point
    3. find the equation of the line normal to an implicitly defined function at a point
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives
 

Section 3.7: Derivatives of Inverse Functions and Logarithms

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. differentiation rules from previous sections
    2. properties of logarithms
  2. procedural knowledge
    1. find the derivative of a function using any previous rule(s)
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives

Learning Goals

  1. declarative knowledge (definitions)
    1. Derivative Rule for Inverses
    2. Derivative of ln(x)
    3. Derivative of logb(x)
    4. Derivative of ax
    5. logarithmic differentiation
  2. procedural knowledge
    1. find the derivative of any order of an function containing logarithms
    2. find the derivative of any order of an function containing exponentials
    3. use logarithmic differentiation when necessary
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives
 

Section 3.8: Inverse Trigonometric Functions

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. differentiation rules from previous sections
  2. procedural knowledge
    1. evaluate inverse trigonometric functions
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives

Learning Goals

  1. declarative knowledge (definitions)
    1. derivative rules for inverse trigonometric functions
  2. procedural knowledge
    1. find the derivative of any order of inverse trigonometric functions
    2. evaluate limits of inverse trigonometric functions
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives
 

Section 3.9: Related Rates

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. differentiation rules from previous sections
    2. various formulas for area, surface area, volume, distance, circumference, and perimeter
    3. definitions of trigonometric ratios on the right triangle
  2. procedural knowledge
    1. find the derivative of a function using any rule(s) from the previous sections
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives

Learning Goals

  1. declarative knowledge (definitions)
    1. none
  2. procedural knowledge
    1. solve problems using related rates
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives
    2. identify a question as one containing related rates
    3. know the general strategy to solve related rates questions
 

Section 3.10: Linearization and Differentials

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. evaluate the trigonometric ratio for any multiple of reference angles
    2. equation of tangent line
    3. differentiation rules from previous sections
  2. procedural knowledge
    1. find the equation of the tangent line to a function at a point
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives

Learning Goals

  1. declarative knowledge (definitions)
    1. linearization, L(x)
    2. standard linear approximation
    3. differential
  2. procedural knowledge
    1. find the linearization of a function at a point
    2. find the differential, dy, of a function
    3. find the change, Δf = f(x0 + dx) – f(x0), of a function
    4. find the value of the estimate, df = f ′(x0)dx, for a function
    5. find the approximation error, |Δf – df|
  3. conditional knowledge
    1. identify the center of the lineariztion that provides the simplest linearization
    2. know when to use the appropriate rules for finding derivatives
 

Section 3.11: Hyperbolic Functions

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. differentiation rules from previous sections
  2. procedural knowledge
    1. find the derivative of a function using any rule(s) from the previous sections
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives

Learning Goals

  1. declarative knowledge (definitions)
    1. hyperbolic functions
    2. identities for hyperbolic functions
    3. differentiation rules for hyperbolic functions
    4. inverse hyperbolic functions
    5. differentiation rules for inverse hyperbolic functions
  2. procedural knowledge
    1. use definitions of sinh(x) and cosh(x) to prove identities for hyperbolic functions
    2. find the derivative of functions containing hyperbolic functions
    3. find the derivative of functions containing inverse hyperbolic functions
  3. conditional knowledge
    1. know when to use the appropriate rules for finding derivatives

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