Quick answer
A derivative describes an instantaneous rate of change: how quickly one quantity is changing with respect to another at a point.
Think about speed
Average speed over a trip uses total distance divided by total time. Instantaneous speed asks how quickly position changes at one moment.
Connect to the slope of a curve
A derivative is the slope of the tangent line at a point, when that derivative exists.
Try a simple example
If f(x) = x², its derivative is f′(x) = 2x. At x = 3, the tangent slope is 6.
Worked examples
f(x)=x² → f′(x)=2xf(x)=3x → f′(x)=3f(x)=5 → f′(x)=0
Common mistake to avoid
A derivative is not the same as the average slope over a wide interval; it measures local change.
Practice questions
Try these without looking at the answers first. Explain your reasoning aloud or write each step.
- Question:
Derivative of 4x— Answer:4 - Question:
Derivative of x² at x=2— Answer:4 - Question:
Derivative of a constant— Answer:0
What to learn next
Review the related lessons linked below, then practice independently. If you make an error, write down which rule you used and why it did not apply.