Basic Derivative Rules

What are the Questions?

Interpret the limit definition of the derivative of a function f(x) at a point x = a as, the slope of the graph of a function f(x) at a point x = a, as the slope of the tangent to the curve at a point x = a, and as the rate of change of a function f(x) with respect to x at x = a.
Recognize and be flexible in the use of different representations of the definition of derivative.
Interpret and be able to represent the definition of derivative analytically, graphically, and numerically.
Interpret the derivative of a function and state the domain for the derivative, namely, identifying the points of a function f for which the derivative does not exist.
Analytically determine the derivative of a function with respect to x at a generic x-value.
Interpret and use (apply?) different notation representing the derivative of a function.
Estimate the derivative of a function f with respect to x at a point (a, f(a) using the graph of the function and the tangent line to the function at x = a. Similarly, graph the function and the tangent line to the function for x = a when given f'(a).
Give examples of functions where the derivative at a point does not exist and why.
Give examples of continuous functions which are not differentiable.
Calculate derivatives of functions (without the use of technology) by applying differentiation rules, including the derivative of: a constant function, a constant multiple of a function, the power of x, a sum of functions.
Relate the acceleration of an object to the object’s position and object’s velocity.
Apply knowledge of the position, velocity, and acceleration functions to solve problems related to motion.
Apply the Chain Rule to find the derivative of a composition of functions. Identify the order at which functions are embedded in another, particularly when the chain rule needs to be repeated to find the derivative of a function.
Apply the chain rule to differentiate implicitly defined functions.
State the definition one-to-one function. State why only a one-to-one function can have an inverse.
Define the inverse sine and cosine functions, including the domain, and range of these functions and their relation to the sine and cosine function. Draw the graphs of the inverse sine and cosine functions, and describe the relation to the graphs of the sine and cosine functions.
Set up and solve related rates problems.
Given a function that is differentiable at a point (c, f(c)) be able do describe what you will see if you zoom in on the graph at this point.