INTODUCTION TO DERIVATIVES

• Secant line


  A line passing through two points of a curve
• Slope of secant line

  (-f(x2) - f(x1)) / (x2 - x1)
• Tangent line


  A straight line that passes through a point on a curve and has the same slope as the curve at that point
• Slope of tangent line

  lim(Δx→0) (f(x0 + Δx) - f(x0)) / Δx, where Δx ≠ 0
• Not all functions have a tangent line at all points
• Function with no tangent line


  • f(x) = |x|
• Derivative


  The limit of the slope of the secant line as the two points approach each other, provided the limit exists
• Computing the derivative


  lim(Δx→0) (f(x + Δx) - f(x)) / Δx
• If a function is differentiable at a point, it is also continuous at that point
• The derivative of f(x) = 3x^2 + 2x - 1 is f'(x) = 9x + 2
• The expression d/dx is called a differential operator and tells you to take the derivative of whatever expression follows