Essential Understanding 2.1

Essential Understanding 2.1 Students will understand that the derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

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Just like limits, you can choose from a variety of strategies to determine the derivative of a function. And just like limits, some strategies produce mathematically perfect answer while others give estimates. And just like limits, you could find derivatives the long way or use some shortcuts. You'll learn all of these strategies and tradeoffs here, because all have their uses. Until you memorize all the shortcuts. This is one time in math that you definitely want to memorize the shortcuts.

Learning Objective 2.1A Shows you the theory of differentiation and the long way of doing it

Learning Objective 2.1B Shows you how to estimate derivatives using tables and graphs

Learning Objective 2.1C Shows you the shortcuts, and the chain rule

Learning Objective 2.1D Shows you second derivatives, third derivatives, and beyond

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Learning Objective 2.1A Students will be able to identify the derivative of a function as the limit of a difference quotient.

Essential Knowledge 2.1A1 Students will know that the difference quotients

and express the average rate of change of a function over an interval.

Essential Knowledge 2.1A2 Students will know that the instantaneous rate of change of a function at a point can be expressed by

or , provided that the limit exists. These are common forms of the definition of the derivative and are denoted f '(a).

Essential Knowledge 2.1A3 Students will know that the derivative of f is the function whose value at x is provided this limit exists.

Essential Knowledge 2.1A4 Students will know that for y = f(x), notations for the derivative include , f '(x), and y'.

Essential Knowledge 2.1A5 Students will know that the derivative can be represented graphically, numerically, analytically, and verbally.

Learning Objective 2.1B Students will be able to estimate derivatives.

Essential Knowledge 2.1B1 Students will know that the derivative at a point can be estimated from information given in tables or graphs.

Learning Objective 2.1C Students will be able to calculate derivatives.

Essential Knowledge 2.1C1 Students will know that direct application of the definition of the derivative can be used to find the derivative for selected functions, including polynomial, power, sine, cosine, exponential, and logarithmic functions.

Essential Knowledge 2.1C2 Students will know that specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric.

Essential Knowledge 2.1C3 Students will know that sums, differences, products, and quotients of functions can be differentiated using derivative rules.

Essential Knowledge 2.1C4 Students will know that the chain rule provides a way to differentiate composite functions.

Essential Knowledge 2.1C5 Students will know that the chain rule is the basis for implicit differentiation.

Essential Knowledge 2.1C6 Students will know that the chain rule can be used to find the derivative of an inverse function, provided the derivative of that function exists.

Learning Objective 2.1D Students will be able to determine higher order derivatives.

Essential Knowledge 2.1D1 Students will know that differentiating f 'produces the second derivative f '', provided the derivative of f ' exists; repeating this process produces higher order derivatives of f.

Essential Knowledge 2.1D2 Students will know that higher order derivatives are represented with a variety of notations. For y = f(x), notations for the second derivative include

, f ''(x), and y''. Higher order derivatives can be denoted or f ⁽ⁿ⁾(x).