The derivative function is so useful that mathematicians invented several ways to write it. They all mean the same thing — "the rate at which the output changes as the input changes" — but each highlights a different angle, and you will meet all of them.

Lagrange's prime notation

The most compact is the prime. If y = f(x), its derivative is written:

Read f'(x) as "f prime of x". It is short and perfect for naming the whole derivative function or evaluating it at a point, like f'(3). Higher derivatives stack more primes: f''(x), f'''(x).

Leibniz's \frac{dy}{dx} notation

Leibniz wrote the derivative as a ratio, echoing the difference quotient \tfrac{\Delta y}{\Delta x} with the step shrunk to nothing:

Read \tfrac{dy}{dx} as "dee y dee x" — the change in y per change in x. The \tfrac{d}{dx} form is an operator: it says "take the derivative with respect to x of whatever follows", as in \tfrac{d}{dx}(x^2) = 2x. This notation shines when you care which variable you are differentiating with respect to, and it makes the chain rule look like cancelling fractions.

Same idea, side by side

Here is one statement written three ways. To evaluate at a point, Leibniz uses a vertical bar:

Step through the table below to line up Lagrange and Leibniz for the same function, the same derivative, and the same value at a point.

The same slope, however you write it

Whatever symbol you use, it names one thing: the slope of the tangent line. Slide x and watch the single number that f'(x), y', and \tfrac{dy}{dx} all report.

Watch it explained

Sal Khan tells the story of Newton's and Leibniz's notations — and why we still use both.