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ScientistGrades 11–12⏱ 20 min

The derivative: slope of a curve

Slide a point along a curve and shrink a secant until it becomes the tangent. Its slope is the derivative.

Standards: HSF-IF.B.6

1Wonder

A car’s speedometer shows 60 km/h at this very instant. But speed is distance divided by time, and an instant has no time at all. How can there be a speed at one moment?

2Explore

Tangent lab

This activity needs a bigger screen

Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.

−4−224−8−448QPy = x²P = (1, 1), h = 1.5Secant slope(f(x + h) − f(x)) / h = 3.5Tangent slope2risingSillyChili
Curve
Try this
  • Drag P along y = x². Where is the tangent slope zero? Where is it negative?
  • Press Shrink h to 0 and watch the blue secant line. What does it turn into?
  • At x = 1, 2 and 3 on y = x², read the tangent slope. Can you guess the rule?
  • Turn on Draw the slope graph for y = x³ − 3x. Where does the slope graph cross zero?

3Explain

2121m = 2SillyChili
Step 1 of 6

A line has one slope

For a straight line, rise over run is the same everywhere. This line goes up 2 for every 1 across: its slope is 2.

The derivativef′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

The derivative is the slope of the tangent: how fast f is changing at the point x.

4Practice

Question 1 of 5

What is the slope of the secant of y = x² from x = 2 to x = 4?

5Recap

  • A secant gives the average slope between two points.
  • Shrinking the gap turns the secant into the tangent.
  • The derivative f′(x) is the tangent’s slope: the rate of change at one point.
  • The derivative of x² is 2x.
  • Peaks and valleys of a smooth curve have derivative 0.