The definition
A derivative is a rate of change. If \(f\) tells you where something is, \(f'\) tells you how fast that is changing — and it does so at every single instant, not averaged over an interval.
The formal definition builds that instant out of intervals that shrink to nothing:
\[f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]
The fraction is the slope of a line through two nearby points on the curve. As \(h\) shrinks those points slide together, and the line settles onto the tangent. The derivative is that limiting slope.
See it on any function
Enter a function and watch the derivative appear beside it, with each rule named in the order the engine applied it.
Three ways to read the same number
The same object answers three questions that look unrelated until you have met all three:
- Geometrically — the slope of the tangent to the curve \(y = f(x)\).
- Physically — an instantaneous rate: position gives velocity, velocity gives acceleration, charge gives current.
- Numerically — a multiplier. Nudge the input by a tiny \(\Delta x\) and the output moves by roughly \(f'(x)\,\Delta x\).
That last reading is the one that makes derivatives useful in error analysis and optimisation, and it is the reason the units of \(f'\) are always "units of \(f\) per unit of \(x\)".
Notation
Four notations are in common use, and you will meet all of them:
- \(f'(x)\) — Lagrange. Compact; higher derivatives become \(f''\), \(f'''\), then \(f^{(4)}\).
- \(\dfrac{dy}{dx}\) — Leibniz. Names both variables, which matters as soon as there is more than one.
- \(\dot{y}\) — Newton. Almost always a derivative with respect to time, common in mechanics.
- \(D_x f\) — operator notation, seen in differential equations.
Leibniz notation is worth being comfortable with, because the chain rule \(\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}\) is nearly self-explanatory when written that way.
Where derivatives appear in each course
Calculus 1 — where derivatives are introduced
Limits, the definition of the derivative, and then the rules that replace it: power, product, quotient and chain, plus the derivatives of the standard functions. The applications are related rates, optimisation, and curve sketching with the first and second derivative tests. Everything on the rules pages belongs here.
Calculus 2 — derivatives as a tool
The focus moves to integration, but derivatives keep working: implicit differentiation of inverse functions, logarithmic differentiation for awkward products and powers, L'Hôpital's rule for indeterminate limits, and the derivatives inside Taylor series. Parametric and polar curves also arrive here in many syllabuses.
Calculus 3 — derivatives in several variables
Partial derivatives, the gradient, directional derivatives, the multivariable chain rule, and second-derivative tests for surfaces. Nothing about single-variable differentiation is discarded — each partial derivative is an ordinary derivative with the other variables held still.
AP Calculus AB and BC
AB covers limits, the definition, all the differentiation rules, implicit differentiation, related rates and optimisation. BC includes everything in AB and adds parametric, polar and vector-valued derivatives, plus series. Neither exam includes partial derivatives, which is a university topic.
So: is a derivative Calculus 1 or Calculus 2?
Calculus 1. Derivatives are introduced and finished there for functions of one variable; Calculus 2 applies them, and Calculus 3 extends them to several variables.
A sensible order to learn it in
- Understand the limit definition well enough to apply it once or twice by hand.
- Learn the rules that make the definition unnecessary in practice — power, product, quotient, chain.
- Learn the standard function derivatives: trigonometric, exponential, logarithmic, and their inverses.
- Practise recognising which rule a problem needs. This, not the algebra, is what exams actually test.
- Read graphs both ways — from \(f\) to \(f'\) and back.
Then work problems until the recognition step is automatic. The practice generator produces a fresh one on demand, filtered to the rule you are drilling.
Questions
What is a derivative in simple terms?
It is how fast something is changing at one instant. On a graph, it is the steepness of the curve at a single point.
What is the definition of a derivative?
\(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\) — the limit of the average rate of change as the interval shrinks to nothing.
Are derivatives Calculus 1 or Calculus 2?
Calculus 1. They are introduced and completed there for single-variable functions. Calculus 2 uses them as a tool, and Calculus 3 extends them to functions of several variables.
Is a partial derivative Calculus 2 or 3?
Calculus 3 in almost every curriculum, since it belongs to multivariable calculus. It does not appear on AP Calculus AB or BC.
What is the difference between a derivative and a differential?
The derivative \(f'(x)\) is a rate. The differential \(df = f'(x)\,dx\) is the small change in \(f\) produced by a small change \(dx\) — the rate multiplied by the nudge.
Why do we need derivatives?
Anywhere a quantity changes: velocity and acceleration in physics, marginal cost in economics, reaction rates in chemistry, and every optimisation problem, since maxima and minima sit where the derivative is zero.