A derivative is a rate of change
For a straight line, the rate is just the slope. For a curve the slope keeps changing, so you ask for the slope at one particular point.
Class 11 to 12 · 62 games
Calculus, explained with pictures
Measuring how fast something is changing at one exact moment.
In everyday life: The speed on a speedometer — not your average speed, your speed right now.
Differentiating a polynomial follows a pattern
Each term x^n becomes n times x^(n-1): the power drops by one and comes down in front. A constant term has no slope, so it disappears.
An average rate is not the rate right now
Over a whole journey you can only work out an average: how far you went divided by how long it took. That is the slope of the straight line joining where you started to where you finished. The speed at one instant is the slope of the curve itself at that single point, and on a journey that is speeding up the two are different numbers.
A worked example
Differentiate x^3 + 5x.
3x^2 + 5
x^3 gives 3x^2, and 5x gives 5. Each power came down in front and dropped by one.
Words to know
- derivative
- The rate at which a quantity is changing.
- tangent
- The straight line that just touches a curve at one point, with the same slope.
- average rate
- The total change divided by the total time — the slope of a straight line between two points.
In the games
You will match functions to their derivatives and find the slope of a curve at a point.
Skills
- Differentiate a polynomial · Class 11
- Limits · Class 11
- Slope of a curve at a point · Class 12
- Continuity at a point · Class 12
- Product and chain rules · Class 12
- Maxima and minima · Class 12
- Integrate a polynomial · Class 12
- The definite integral as area · Class 12
- Solve a simple differential equation · Class 12
- Average rate and rate at an instant · Class 12
- What f′ says about the shape of f · Class 12
- Optimisation — the biggest box from a sheet · Class 12
By class
Class 11 calculus (8) · Class 12 calculus (54)
Practise calculus in Tivumo
No download, no account for your child. Classes 3 to 12, more than 1,000 games.