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Unit 2: Differentiation: Definition and Fundamental Properties

Syllabus
2020
Section
—
Level
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Exam analysis

2.1 Defining Average and Instantaneous Rates of Change at a Point

Syllabus
2020
Topic
2.1
Level
—

CHA-2.A—Determine average rates of change using difference quotients

  • CHA-2.A Determine average rates of change using difference quotients.
  • CHA-2.A.1 The difference quotients (f(a + h) − f(a))/h and (f(x) − f(a))/(x − a) express the average rate of change of a function over an interval.
  • Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

CHA-2.B—Represent the derivative of a function as the limit of a difference quotient

  • CHA-2.B Represent the derivative of a function as the limit of a difference quotient.
  • CHA-2.B.1 The instantaneous rate of change of a function at x = a can be expressed by lim h→0 (f(a + h) − f(a))/h or lim x→a (f(x) − f(a))/(x − a), provided the limit exists. These are equivalent forms of the definition of the derivative and are denoted f′(a).
  • Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

2.2 Defining the Derivative of a Function and Using Derivative Notation

Syllabus
2020
Topic
2.2
Level
—

CHA-2.B—Represent the derivative of a function as the limit of a difference quotient—Topic 2.2

  • CHA-2.B Represent the derivative of a function as the limit of a difference quotient.
  • CHA-2.B.2 The derivative of f is the function whose value at x is lim h→0 (f(x + h) − f(x))/h, provided this limit exists.
  • CHA-2.B.3 For y = f(x), notations for the derivative include dy/dx, f′(x), and y′.
  • CHA-2.B.4 The derivative can be represented graphically, numerically, analytically, and verbally.
  • Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

CHA-2.C—Determine the equation of a line tangent to a curve at a given point

  • CHA-2.C Determine the equation of a line tangent to a curve at a given point.
  • CHA-2.C.1 The derivative of a function at a point is the slope of the line tangent to a graph of the function at that point.
  • Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

2.3 Estimating Derivatives of a Function at a Point

Syllabus
2020
Topic
2.3
Level
—

CHA-2.D—Estimate derivatives

  • CHA-2.D Estimate derivatives.
  • CHA-2.D.1 The derivative at a point can be estimated from information given in tables or graphs.
  • CHA-2.D.2 Technology can be used to calculate or estimate the value of a derivative of a function at a point.
  • Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

Syllabus
2020
Topic
2.4
Level
—

FUN-2.A—Explain the relationship between differentiability and continuity

  • FUN-2.A Explain the relationship between differentiability and continuity.
  • FUN-2.A.1 If a function is differentiable at a point, then it is continuous at that point. In particular, if a point is not in the domain of f, then it is not in the domain of f '.
  • FUN-2.A.2 A continuous function may fail to be differentiable at a point in its domain.
  • Enduring understanding FUN-2: Recognizing that a function’s derivative may also be a function allows us to develop knowledge about the related behaviors of both.

2.5 Applying the Power Rule

Syllabus
2020
Topic
2.5
Level
—

FUN-3.A—Calculate derivatives of familiar functions

  • FUN-3.A Calculate derivatives of familiar functions.
  • FUN-3.A.1 Direct application of the definition of the derivative and specific rules can be used to calculate the derivative for functions of the form f(x) = xʳ.
  • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

Syllabus
2020
Topic
2.6
Level
—

FUN-3.A—Calculate derivatives of familiar functions—Topic 2.6

  • FUN-3.A Calculate derivatives of familiar functions.
  • FUN-3.A.2 Sums, differences, and constant multiples of functions can be differentiated using derivative rules.
  • FUN-3.A.3 The power rule combined with sum, difference, and constant multiple properties can be used to find the derivatives for polynomial functions.
  • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

2.7 Derivatives of cos x, sin x, eˣ, and ln x

Syllabus
2020
Topic
2.7
Level
—

FUN-3.A—Calculate derivatives of familiar functions—Topic 2.7

  • FUN-3.A Calculate derivatives of familiar functions.
  • FUN-3.A.4 Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions.
  • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

LIM-3.A—Interpret a limit as a definition of a derivative

  • LIM-3.A Interpret a limit as a definition of a derivative.
  • LIM-3.A.1 In some cases, recognizing an expression for the definition of the derivative of a function whose derivative is known offers a strategy for determining a limit.
  • Enduring understanding LIM-3: Reasoning with definitions, theorems, and properties can be used to determine a limit.

2.8 The Product Rule

Syllabus
2020
Topic
2.8
Level
—

FUN-3.B—Calculate derivatives of products and quotients of differentiable functions

  • FUN-3.B Calculate derivatives of products and quotients of differentiable functions.
  • FUN-3.B.1 Derivatives of products of differentiable functions can be found using the product rule.
  • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

2.9 The Quotient Rule

Syllabus
2020
Topic
2.9
Level
—

FUN-3.B—Calculate derivatives of products and quotients of differentiable functions—Topic 2.9

  • FUN-3.B Calculate derivatives of products and quotients of differentiable functions.
  • FUN-3.B.2 Derivatives of quotients of differentiable functions can be found using the quotient rule.
  • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Syllabus
2020
Topic
2.10
Level
—

FUN-3.B—Calculate derivatives of products and quotients of differentiable functions—Topic 2.10

  • FUN-3.B Calculate derivatives of products and quotients of differentiable functions.
  • FUN-3.B.3 Rearranging tangent, cotangent, secant, and cosecant functions using identities allows differentiation using derivative rules.
  • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.