Introduction to Derivatives · Grade 12

What 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.

Learning objectives

From chord to tangent

Fix one point on the graph and let the other slide towards it. Each position gives a chord with its own slope, and those slopes settle towards a single number.

That number is the derivative, and the line through the fixed point with that slope is the tangent. The word limit is doing the work here: the second point never actually arrives, because if it did the division would be zero over zero.

This is why the derivative is a statement about one point while the average rate of change is a statement about an interval. One describes the direction of the curve; the other summarises a stretch of it.

Computing from the definition

Write the average rate of change between x and a nearby point a small step h away, simplify the expression until the h in the denominator cancels, and only then let h go to zero.

The cancelling is the whole trick. Before it, substituting zero gives nothing usable; after it, substituting zero is straightforward. Any expression that still has h underneath has not been simplified enough.

What the sign tells you

A positive derivative means the function is rising at that point, and a negative one means it is falling. The size says how steeply.

A derivative of zero means the tangent is horizontal. That is where maxima, minima and points of inflection live, which is the subject of the last lesson in this sequence.

Worked examples

  1. Compute the derivative of f(x) = x squared at x = 3 from the definition

    1. Average rate between 3 and 3 plus h: (3 + h) squared minus 9, over h
    2. Expand the top: 9 plus 6h plus h squared, minus 9
    3. That leaves 6h plus h squared, over h, which cancels to 6 plus h
    4. Let h go to zero

    Answer: The derivative at x = 3 is 6

  2. What is the derivative of a constant function, and why?

    1. The function value is the same at every point
    2. So the change in the function is always zero
    3. Zero divided by any h is zero

    Answer: Zero. A horizontal line has slope zero everywhere

  3. The derivative of a function at a point is minus 4. What does that say?

    1. The sign is negative, so the function is falling there
    2. The size is 4, so it falls steeply
    3. The tangent at that point has slope minus 4

    Answer: It is decreasing at that point, at a rate of 4 units per unit of x

Common mistakes

Substituting h equals zero before simplifying
That gives zero over zero, which says nothing. The h in the denominator has to cancel first, and if it will not cancel the expression has not been expanded far enough.
Treating the derivative as a rate over an interval
The derivative belongs to a single point. An average rate of change belongs to an interval, and the two agree only by coincidence.
Reading a zero derivative as a maximum
A zero derivative means a horizontal tangent, which could be a maximum, a minimum, or neither. Deciding which requires the sign on either side.

What to remember

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