# Unit 2: Differentiation: Definition and Fundamental Properties

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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.

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 Topic Name & Learning Objective Essential Knowledge 2.1 Defining Average Value and Instantaneous Rates of Change at a Point LEARNING OBJECTIVE CHA-2.A Determine average rates of change using difference quotients. CHA-2.A.1 The difference quotients   and   express the average rate of change of a function over an interval. CHA-2.B.1 The instantaneous rate of change of a function at x = a can be expressed by   or   provided the limit exists. These are equivalent forms of the definition of the derivative and are denoted

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Local Linearity 1 The graphical manifestation of differentiability with pathological examples.

Local Linearity 2  Using local linearity to approximate the tangent line. A calculator exploration. Discovering the Derivative   A graphing calculator exploration

The Derivative 1   Definition of the derivative.

The Derivative 2   Calculators and difference quotients

Difference Quotient

Difference Quotients I

Difference Quotients II

At Just the Right Time  A good problem

Tangents and Slopes

What’s a Mean Old Average Anyway?   Helping students understand the difference between the average rate of change of a  function, the average value of a function, and the Mean Value theorem

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 2.2 Defining the Derivative of a Function and Using Derivative Notation LEARNING OBJECTIVES CHA-2.B Represent the derivative of a function as the limit of a difference quotient. CHA-2.C Determine the equation of a line tangent to a curve at a given point. The derivative of f is the function whose value at x is , provided this limit exists. CHA-2.B.3 For , notations for the derivative include , ,  and . CHA-2.B.4 The derivative can be represented graphically, numerically, analytically, and verballHA 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.

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The Derivative I

The Derivative II

Derivative Rules I Using the limit definition

Units  The units of derivatives and integrals

Did He or Didn’t He?  How Fermat found extreme values without derivatives.

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 2.3 Estimating Derivatives of a Function at a Point LEARNING OBJECTIVE 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.

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Derivative Practice – Numbers   Derivative from tables of numbers.

Derivative Practice – Graphs   Derivative from graphs

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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.

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 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist LEARNING OBJECTIVE 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 f ‘. FUN-2.A.2 A continuous function may fail to be differentiable at a point in its domain.

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Differentiability Implies Continuity

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ENDURING UNDERSTANDING
FUN-3 Recognizing opportunities to apply derivative rules can simplify differentiation.

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 2.5 Applying the Power Rule LEARNING OBJECTIVE 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

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Differentiation Techniques  The definition, basics, product rule, Quotient Rule

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 2.6 Derivative Rules: Constant, Sum. Difference, and Constant Multiple LEARNING OBJECTIVE 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.

Blog Posts:

Differentiation Techniques  The definition, basics, product rule, Quotient Rule

The Derivative Rules 1 (9-14-2012) Constants, sums and differences, powers.

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ENDURING UNDERSTANDINGS
FUN-3 Recognizing opportunities to apply derivative rules can simplify differentiation.

LIM-3 Reasoning with definitions, theorems, and properties can be used to determine a limit.

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 2.7 Derivatives of cos x, sin x, ex, and ln x LEARNING OBJECTIVES FUN-3.A Calculate derivatives of familiar functions. LIM-3.A Interpret a limit as a definition of a derivative. FUN-3.A.4 Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions. 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.

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 2.8 The Product Rule LEARNING OBJECTIVE 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.

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Derivative Rules II  the Product Rule

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 2.9 The Quotient Rule LEARNING OBJECTIVE 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.

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Derivative Rules III  the Quotient Rule

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

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