Grade 12

Introduction to Derivatives

The derivative built up from average rate of change, then the differentiation rules, the tangent line, and the first stage of studying a function.

6 lessons

  1. 1What is an average rate of change and how does it relate to the slope of a chord?An average rate of change is the change in the function divided by the change in x over an interval. Geometrically it is the slope of the chord joining the two points on the graph — the straight line through them — which is why the same number answers both a question about speed and a question about slope.35m
  2. 2What is the derivative of a function at a point?The derivative at a point is the limit of the average rate of change as the interval shrinks to nothing. Geometrically it is the slope of the tangent to the graph at that point — the single straight line that matches the direction of the curve there rather than cutting across it.45m
  3. 3How do you differentiate a polynomial?Use the power rule on each term: bring the exponent down in front as a multiplier and reduce the exponent by one. A coefficient comes along unchanged, a sum is differentiated term by term, and a constant term differentiates to zero. The result is a polynomial one degree lower.40m
  4. 4When do you use the product, quotient and chain rules?Look at how the function is built. Two things multiplied need the product rule, one thing divided by another needs the quotient rule, and a function sitting inside another function needs the chain rule. Identifying the structure correctly is most of the work; the formulas themselves are short.45m
  5. 5How do you find the equation of a tangent to a graph?Three steps. Find the y-coordinate of the point by substituting into the function, find the slope by substituting into the derivative, then put the point and the slope into the equation of a straight line. The order matters because the derivative gives a slope, never a point.35m
  6. 6How do you find intervals of increase and decrease and points of extremum?Differentiate, set the derivative equal to zero and solve to find the candidate points, then check the sign of the derivative on either side of each. Positive means increasing, negative means decreasing, and a change of sign at a point makes it a maximum or a minimum depending on the direction of the change.45m