What is a derivative? Explained step by step

The idea: a rate of change

A function says how one quantity depends on another: distance on time, temperature on the hour, cost on the number of items. The average rate of change over a stretch is easy: the change in $y$ divided by the change in $x$.

The rate at a single point is harder. In a single instant the car covers no distance, so “distance over time” gives $\frac{0}{0}$. The derivative gets round this: we work out the average rate over shorter and shorter stretches and watch which number it closes in on.

The same idea turns up everywhere:

• speed is the derivative of distance with respect to time, and acceleration is the derivative of speed,
• how fast something warms up is the derivative of its temperature,
• in economics, marginal cost says how much more it costs to make one more item.

From a secant to the tangent

On a graph, the average rate of change is the slope of a secant - a line through two points of the graph. Go through the animation step by step with the buttons or the arrow keys.

Notice that we never put $h = 0$ - that would give $\frac{0}{0}$. We only look at what the secant’s slope gets close to. That is what a limit is.

Try it yourself: the derivative at every point

A derivative is not just one number. At every point of the graph the tangent has a different slope. Move the point $x_0$ and watch the tangent. Every position leaves a dot on the lower graph, at the height of the tangent’s slope. Before you show the whole graph of $f'$, try to guess what it looks like.

What to notice:

• for $f(x) = x^2$ the dots line up into the straight line $y = 2x$ - the graph of the derivative,
• where the function rises the derivative is positive, and where it falls the derivative is negative,
• at a peak or a dip the tangent is flat, so the derivative is zero; for $x^3 - 3x$ that happens at $x = -1$ and $x = 1$,
• the graph of the derivative of $\sin x$ is the graph of $\cos x$.

The definition

The derivative of a function $f$ at a point $x_0$ is the limit

provided the limit exists and is finite. The fraction inside the limit is the difference quotient - the slope of the secant from the animation. Working out the derivative at every point gives a new function, $f'(x)$: it gives the slope of the tangent for every $x$.

The tangent to the graph at the point $(x_0, f(x_0))$ has slope $f'(x_0)$, so its equation is

Example 1: the derivative of x² from the definition

We work out the difference quotient at any point $x$ and only at the very end let $h$ go to zero:

So $(x^2)' = 2x$. At $x = 1$ that gives $2$ - just what the animation showed.

Example 2: the tangent to x³ at 2

The tangent to $y = x^3$ at the point $(2, 8)$ has slope $12$, and its equation is $y = 12(x - 2) + 8$, that is $y = 12x - 16$.

Your turn

Use the definition to find the derivative of $f(x) = 3x^2 - 2x$ at $x_0 = 1$.

Common mistakes

• Putting $h = 0$ straight away. That gives $\frac{0}{0}$, which is not a number. Simplify the quotient first (cancel the $h$), and only then look at what it tends to.
• Mixing up $f'(x_0)$ and $f'(x)$. $f'(1) = 2$ is a number - the slope at one point. $f'(x) = 2x$ is a function - the slope at every point.
• “A tangent touches the graph at only one point.” Not always. The tangent to $y = x^3$ at $(1, 1)$ is $y = 3x - 2$, and it crosses the graph again at $(-2, -8)$. A tangent is about the direction of the graph at one place, not about how many points they share.
• “Every function has a derivative everywhere.” The graph of $|x|$ has a corner at zero: secants on the right have slope $1$ and on the left $-1$. There is no limit, so there is no derivative at zero.

In exams

In Poland, derivatives are part of the extended-level maths matura: students must work them out (including with the sum, product and quotient rules), use them to find where a function increases or decreases, and solve optimisation problems - the largest area, the lowest cost. In 2026, students scored 47% of the points on average on the task that combined a derivative with optimisation (CKE report, in Polish). At university, derivatives are one of the main topics of first-year calculus, together with the rules of differentiation, L’Hôpital’s rule and curve sketching.

Check yourself

Flashcards for review

Flip a card and say whether you knew the answer. The ones you did not know come back at the end of the deck. Why it pays to come back to them in a few days is explained in the guide Spaced repetition.