The Derivative
The derivative measures instantaneous rate of change: the slope of the tangent.
About this topic
The derivative measures the instantaneous rate of change of a function, the slope of its tangent line at a point. It is defined as the limit of the difference quotient: f′(x) = lim h→0 (f(x + h) − f(x))/h.
Velocity is the derivative of position and acceleration is the derivative of velocity. Geometrically, the slope of the secant line between two points becomes the slope of the tangent line as the points merge. The derivative is itself a function that gives the slope at every x, and it can be written f′(x), dy/dx or df/dx.
Example: for f(x) = x², the difference quotient is ((x + h)² − x²)/h = (2xh + h²)/h = 2x + h, which tends to 2x. So at x = 3 the slope is 6. If a ball is at height 5t² metres after t seconds, its instantaneous speed at t = 2 is 20 m/s.
Derivatives help find maximum profit, fastest growth, sharpest curve and best fit in science, economics and engineering. A function must be continuous to be differentiable, but the reverse is false: the corner of the absolute value graph has no derivative. Tip: a derivative of zero means a horizontal tangent, not a zero function.
Key ideas
- f′(x) = lim (f(x+h) − f(x))/h
- Drag the point to see the tangent slope
- Units are y per x
Quick check
Use cases: where The Derivative helps
- Homework: review the key idea behind The Derivative before you start a problem set.
- Exam revision: use the quick check to test whether you can recall The Derivative without notes.
- Visual learning: play with the interactive lab to see how The Derivative changes when you adjust its inputs.
- Teaching a friend: explain The Derivative in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is The Derivative?
The derivative measures instantaneous rate of change: the slope of the tangent.
What will I learn in The Derivative?
f′(x) = lim (f(x+h) − f(x))/h. Drag the point to see the tangent slope. Units are y per x.
Is The Derivative part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the The Derivative lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study The Derivative?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.