- You should also know the relationship between differentiability and continuity. That is, if a
function is differentiable at a point, it’s continuous there. But if a function is continuous at a
point, it’s not necessarily differentiable there.
B.Derivative at a Point
- You should know the Power Rule, the Product Rule, the Quotient Rule, and the Chain Rule.
- You should be able to find the slope of a curve at a point, and the tangent and normal lines to a
curve at a point. - You should also be able to use local linear approximation and differentials to estimate the
tangent line to a curve at a point. - You should be able to find the instantaneous rate of change of a function using the derivative or
the limit of the average rate of change of a function. - You should be able to approximate the rate of change of a function from a graph or from a table
of values. - You should be able to find Higher-Order Derivatives and to use Implicit Differentiation.
C.Derivative of a Function
- You should be able to relate the graph of a function to the graph of its derivative, and vice-
versa. - You should know the relationship between the sign of a derivative and whether the function is
increasing or decreasing (positive derivative means increasing; negative means decreasing). - You should know how to find relative and absolute maxima and minima.
- You should know the Mean Value Theorem for derivatives and Rolle’s Theorem.
D.Second Derivative
- You should be able to relate the graph of a function to the graph of its derivative and its second
derivative, and vice-versa. This is tricky. - You should know the relationship between concavity and the sign of the second derivative
(positive means concave up; negative means concave down).