Differentiation

• Differentiation


  A process that helps us to calculate gradient or slope of a function at different points. It also helps us to identify change in one variable with respect to another variable.
• Derivative


  Notation:

  dy/dx of f(x) represents derivative of y = f(x) with respect to x
• Gradient of curves


  • Unlike straight lines, the gradient of a curve changes constantly. The gradient of a curve at any given point is the same as calculating gradient of the tangent at that given point. You can find exact gradient of a curve at any given point using derivatives.
• Finding derivative


  Learn how to find derivative of a function (i.e. exact gradient of a curve or function at a given point)
• f(x) = x^2


  • Show that f'(x) = lim h→0 (2x + h)

  b. Hence deduce that f'(x) = 2x
• Differentiating x^n
• Differentiating ax^n
• f(x)


  • x^3

  b. 10x^-1
• Gradient of a curve at any given point


  • Calculating gradient of the tangent at that given point

  Finding derivative at that given point
• Differentiating quadratics
• y = 2x^2 - x - 1


  • Find the gradient of the curve at the point (2,5)
• Equation of tangent


  y - f(a) = f'(a)(x - a)
• Equation of normal


  y - f(a) = -1/f'(a)(x - a)
• Increasing and decreasing function

  • Increasing if f'(x) ≥ 0

  Decreasing if f'(x) ≤ 0
• Second order derivative


  When you differentiate a function f(x) once it's called first order derivative f'(x)

  When you differentiate a function f(x) twice, it's called second order derivative f''(x)
• y = 2x^2 + 7x - 3


  • Find f'(x) and f''(x)
• Stationary point


  A point on the curve where the gradient of the curve is 0
• Determining nature of stationary points
• Point (x1, x2)


  Given by y - y1 = m(x - x1)
• First order derivative


  When you differentiate a function y = f(x) once, it is called the first order derivative and denoted as f'(x)
• Second order derivative


  When you differentiate a function f(x) twice, it is called the second order derivative and denoted as f''(x)
• Finding first and second order derivatives

  1. Differentiate y = 2x^2 + 7x - 3

  2. f'(x) = 4x + 7

  3. f''(x) = 4
• Stationary point


  Any point on the curve where the gradient of the curve is 0
• Types of stationary points


  • Local maximum
  • Local minimum
  • Point of inflection
• Determining nature of stationary point using second order derivative


  • If f''(a) > 0 then local minimum
  • If f''(a) < 0 then local maximum
  • If f''(a) = 0 then could be local minimum, local maximum or point of inflection
• Finding stationary point of y = x^4 - 32x


  1. Find derivative and set equal to 0

  2. 4x^3 - 32 = 0

  3. x = 2

  4. Substitute x = 2 into original equation to get y = -48

  5. Stationary point is (2, -48)
• Derivative


  Represents rate of change of y with respect to x
• Normal to a curve

  A straight line passing through a point on the curve and perpendicular to the tangent line at that point
• Finding equation of normal to curve y = f(x) at point (x1, f(x1))


  1. Gradient of normal = -1/f'(x1)

  2. Equation of normal is y = -(1/f'(x1))(x - x1) + f(x1)

  
• The gradient of perpendicular lines are negative reciprocals of each other
• Differentiating from first principles: f'(x) = lim(h->0) (f(x+h) - f(x))/h