Derivatives

Derivatives

1. Differntiability

   • How f´(a) Might Fail to Exist
   • Differntiability Implies Local Linearity
     • Locally linear
   • Numerical Derivatives on a Calculator
     • Symmetric difference quotient
     • Numeriacal Derivative
       • Numerical derivative of f at a
       • Numerical derivative of f
   • Differentiability Implies Continuity
   • Intermediate Value Theorem for derivatives

2. Rules for Differentiation

   • Positive Integer Powers, Multiples, sums, and differences
     • Rule 1: Derivative of a Constant Function
     • Rule 2: Power Rule for Positive Integer Powers of x
     • Rule 3: The Constant Multiple Rule
     • Rule 4: The Sum and Difference Rule
   • Products and Quotients
     • Rule 5: The Product Rule
     • Rule 6: The Quotient Rule
   • Negative Integer Powers of x
     • Rule 7: Power Rule for Negative Integer Powers of x
   • Second and Higher Order Derivatives
     • nth derivative

3. Velocity and Other Rates of Change

   • Instantaneous Rates of Change
     • Instantaneous rate of Change
   • Motion Along a Line
     • Displacement
     • Instantaneous Velocity
     • Speed
     • Acceleration
   • Sensitivity to Change
     • Sensitive
   • Derivatives in Economics
     • Marginal Cost
     • Marginal Revenue

4. Derivative of a Function

   • Definition of a derivative
     • derivative of f at a
     • Derivative
       • If f(x) exists, we say that f has a derivative(is differentiable)
       • A function that is differntiable at every point of its domain is a differentiable function.
     • Derivative at a point
       • F at the point of x= a
   • Notation
   • Relationship between the graphs f and f´
   • Graphing the Derivative from Data
   • One-sided Derivatives
     • Differentiable on a closed interval [a,b]
     • Right-hand derivative at a
     • left-hand derivative at b

5. Derivatives of Trigonometric Functions

   • Derivative of the Sine Function
   • Derivative of the Cosine Function
   • Simple Harmonic Motion
   • Jerk
   • Derivatives of the Other Basic Trigonometric Functions