Derivative from first principles

Slide a second point towards P and watch the secant slope settle on the tangent slope. About 8 minutes, GCSE to A-level.

What you will do

1. How steep is a curve?. A straight line has one slope. A curve is different: its steepness changes from place to place. Here is f(x) = x^2 and a point P on it at x = px. Our question: how steep is the curve at P?
2. Use a second point. We can only find the slope of a line easily, so pick a second point Q a small distance h to the right of P and draw the line through both. Its slope is rise over run: f(x+h)-f(x)h. Here h = h, so Q is at x = 3.
3. Slide Q towards P. Now make h smaller. Drag the h slider (or press play) until it is below 0.1 and watch the line pivot while the table fills in.
4. What are the slopes heading for?. Look at the table: as h shrinks the slopes from the right and from the left squeeze together. Their common limit is the slope of the tangent — the derivative.
5. The derivative function. Letting h 0 gives the tangent and its slope f'(x). Now we plot that slope for every x as a second curve (dashed). The marker sits at height slope above P.
6. Move P. The marker on the dashed curve always sits directly above or below P at the height of the slope. Drag P along the curve to x = 2.
7. Predict the slope. At x = 2 the slope reads about slope. Notice the pattern: the slope at x looks like twice x.
8. From numbers to the rule. Open the Maths tab: the engine derived f'(x) using the power rule and then checked it against a numeric difference quotient at dozens of points. The rule and the table agree.
9. Your turn. Everything is unlocked. Type your own function (try x^3 - 3x or x), drag P, use secant mode, or switch to Area to see the reverse process, integration.