Intro to Derivatives

• Difference quotient

  The expression (f(a+h) - f(a)) / h
• Instantaneous rate of change at x=a

  The limit of the difference quotient as h approaches 0
• Finding the slope of the tangent to f(x) = -x^2 + x at x = -1


  1. Use the difference quotient

  2. Substitute values

  3. Simplify
• First principles definition of the derivative

  f'(x) = lim (h->0) (f(x+h) - f(x)) / h, if the limit exists
• Differentiation

  The process of finding the slope of the tangent to a curve
• Derivative

  An expression that allows you to calculate the slope of the tangent at any given point on the graph of the function
• Finding the slope of the tangent to f(x) = x^3 + 4 at x = 2 and x = -1 using first principles

  1. Apply first principles definition

  2. Substitute values

  3. Simplify
• Finding the equation of the tangent to f(x) = 1/x at the point P(1/2, 2)

  1. Apply first principles definition

  2. Substitute values

  3. Simplify

  4. Find equation of tangent line
• Notations for the derivative


  • y' = f'(x) = dy/dx = d/dx f(x)
  • The value of the derivative when x=a can be written as (dy/dx)_x=a
• The derivative of a function y = f(x) is f'(x) = lim (h->0) (f(x+h) - f(x)) / h
• The derivative represents the slope of the tangent to y = f(x) at a point P(x, f(x))
• The process of finding the slope of a tangent to a curve is called differentiation
• The derivative only exists if the limit exists
• Scenarios where derivatives fail to exist

  • Discontinuity
  • Sharp point
  • Vertical inflection point
  • Corner/cusp