Rates of Change & the Derivative

Average speed over a shorter and shorter trip settles on the speed at one instant.

A car covers 60 miles in one hour. What was its speed?

On average, 60 mph. But the speedometer was never fixed at 60 — it rose, fell, and stopped at lights.

average speed, over shorter and shorter trips bring the second point closer and the slope settles on one number

Ask about a shorter stretch and the average gets closer to the truth. One minute is better than one hour.

One second is better still. Keep shrinking the interval and the averages settle on a single number.

That number is the speed at that instant, and it is what the speedometer reads. A derivative is exactly this: the slope between two points, as the points close together.

The tangent line, and the slope it carries run rise slope here = the rate right now one line touching at one point

Average rate of change is just slope

Between two points on any graph, the average rate of change is f(b)-f(a)b-a.

That is rise over run, which you have been computing since Grade 8. Nothing new has happened yet.

On a distance-time graph it is average speed; on a cost graph it is cost per unit.

Units come free

The units of a rate are always the y-units over the x-units: miles per hour, dollars per item, degrees per minute.

Shrinking the gap

Now hold the first point still and slide the second one towards it. The secant line pivots.

As the gap closes, the secant settles onto the tangent — the line with the limiting secant slope at that point. It may cross the curve or meet it again elsewhere.

The slope of that tangent is the instantaneous rate of change.

Why not just set the gap to zero

Then the slope reads 00, which says nothing. The limit is what gets you past that, and it is why the previous unit came first.

The difference quotient

Write the gap as h. The slope over that gap is f(x+h)-f(x)h.

Simplify until the h in the denominator cancels, then let h → 0.

For f(x)=x2: the quotient is 2xh+h2h = 2x+h, which goes to 2x.

Cancel before you take the limit

Setting h=0 too early gives 00. The algebra has to remove the h from the bottom first, for these polynomial examples. Some functions are not differentiable: |x| at zero has different left and right slopes.

The power rule

Do the difference quotient a few times and a pattern appears: the derivative of xn is nxn-1.

Bring the power down in front, and drop it by one. That is the rule, and it covers most of what you meet.

A constant has derivative zero, because a horizontal line has slope zero everywhere.

Check it against a line

For f(x)=5x, the rule gives 5 — and a straight line of slope 5 does have slope 5 everywhere. The rule agrees with what you already knew.

What the sign tells you

A positive derivative means the function is rising; negative means falling.

At a differentiable interior maximum or minimum the derivative is zero, but a zero derivative need not be an extremum. Endpoints and corners need separate checks.

So the derivative turns questions about shape into questions about sign.

Zero slope, but which kind

A flat tangent could be a peak, a valley, or a momentary pause. Check whether the slope changes sign around it.

Worked examples

Average rate of change

For f(x)=x2, find the average rate of change from x=1 to x=4.

  1. f(1)=1 and f(4)=16. Evaluate at both ends.
  2. Slope = 16-14-1 = 153. Rise over run.
  3. = 5. The curve climbs an average of 5 per unit across that stretch.

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From the definition

Use the difference quotient to find f'(x) for f(x)=x2.

  1. (x+h)2 - x2h = x2+2xh+h2-x2h. Expand the bracket.
  2. The x2 terms cancel: 2xh+h2h = 2x+h. Now h divides out.
  3. Let h → 0: the derivative is 2x. Matches the power rule.

f'(x)=2x

Using the power rule

Find the slope of y = x3 - 6x at x=2.

  1. Differentiate term by term: dydx = 3x2 - 6. Power down in front, drop it by one.
  2. At x=2: 3(4) - 6 = 6. Substitute after differentiating, never before.
  3. So the tangent there has slope 6. Positive, so the curve is rising at that point.

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Words to know

Average rate of change
The slope between two points, f(b)-f(a)b-a.
Secant line
A line through two points on a curve.
Tangent line
A line touching the curve at one point, with the same direction there.
Derivative
The slope of the tangent, written f'(x) or dydx.
Difference quotient
f(x+h)-f(x)h, the slope over a gap of h.

Mistakes to watch for

Substituting the point before differentiating. For y = x3-6x at x=2, plugging in first gives the number -4, whose derivative is zero. Differentiate first, then substitute.

Setting h=0 too early in the difference quotient. That gives 00 and stops the problem. The algebra must cancel the h from the denominator first.

Confusing average and instantaneous rates. Covering 60 miles in an hour is an average; the speedometer reading is a derivative, and the two agree only by coincidence.

Mishandling the constant. The derivative of 7 is 0, not 7 — a horizontal line has slope zero everywhere.

Dropping the power instead of the exponent. The derivative of x5 is 5x4: the 5 comes down in front and the exponent drops by one. Doing only half gives x4 or 5x5, both wrong.

The speedometer in a car is a physical derivative. It has no way to know your average speed and no interest in it.

Newton called it a fluxion and Leibniz called it a differential. Leibniz’s notation dydx won, because it makes the chain rule look obvious.

The same idea runs everything: marginal cost in economics, reaction rate in chemistry, and growth rate in biology are all derivatives.

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