QuestionQuestionConceptConceptDocumentDocument

Unit 2: Differentiation: Definition and Fundamental Properties

Start with Concept to understand a topic, then use Question Bank to check what you know.

10 topics · 13 learning objectives

Your progress

Sign in to see your mastery and mistakes.

  1. 2.1 Defining Average and Instantaneous Rates of Change at a Point

    1. 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.

    2. 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. 2.2 Defining the Derivative of a Function and Using Derivative Notation

    1. 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.

    2. 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.

  3. 2.3 Estimating Derivatives of a Function at a Point

    1. 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.

  4. 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

    1. 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.

  5. 2.5 Applying the Power Rule

    1. 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.

  6. 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

    1. 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.

  7. 2.7 Derivatives of cos x, sin x, eˣ, and ln x

    1. 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.

    2. 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.

  8. 2.8 The Product Rule

    1. 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.

  9. 2.9 The Quotient Rule

    1. 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.

  10. 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

    1. 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.