Derivative Notation and Rates of Change

The derivative goes by four names — \(f'(x)\), \(y'\), \(\frac{dy}{dx}\), and \(\frac{d}{dx}[f]\) — and in the real world it's a rate of change whose units are the units of \(f\) per unit of \(x\): velocity is the derivative of position.

By the end you'll be able to read all four notations as names for the same derivative, and state what any derivative means in context — with the right units attached.

Predict: the car below has position \(s(t) = t^2\) meters. What should the velocity — the tangent slope, in meters per second — be at t = 3? Decide first, then drag t there and read \(\frac{ds}{dt}\) off the diagram.

The dashed tangent line rides along the position curve as you drag. Its slope is the derivative — the car's velocity right now — and the notation buttons relabel that same slope value with each of its four names. Hover or focus the point (or anywhere on the curve) for exact readouts.

Write the slope as…

slope = ds/dt = 2t = 2.0 meters per second — the velocity at t = 1.0 s

s(t) = t² (meters) — position curve with the tangent at the current time
position s(t) tangent — slope = velocity

One derivative, many costumes: \(f'(x)\), \(y'\), \(\frac{dy}{dx}\), and \(\frac{d}{dx}[f]\) all name the difference-quotient limit from last lesson — and in applications that number is a rate of change carrying units.

Formal

The derivative you defined as a limit last lesson has several standard notations. Lagrange notation writes \(f'(x)\) — or just \(y'\) when the function is called y. Leibniz notation writes \(\frac{dy}{dx}\) or \(\frac{d}{dx}[f(x)]\), naming both the output and the input variable. The symbol \(\frac{dy}{dx}\) is a single object, not a literal quotient — though it behaves like the limit of \(\frac{\Delta y}{\Delta x}\), which is exactly what it abbreviates. Physics adds Newton's dot (\(\dot{y}\)) for time derivatives. To evaluate at a point in Leibniz style, write \(\left.\frac{dy}{dx}\right|_{x=a}\).

Applied

Because the derivative is a limit of \(\frac{\Delta y}{\Delta x}\), it inherits units: the units of the output \(f\) per unit of the input \(x\). That's what makes it physically meaningful. If \(s(t)\) is position in meters and t is seconds, \(s'(t)\) is velocity in meters per second — the tangent slope you just read off the diagram. If \(C(q)\) is the cost in dollars to make \(q\) items, \(C'(q)\) is the marginal cost in dollars per item. If \(T(x)\) is temperature in degrees along a rod measured in cm, \(T'(x)\) is in degrees per cm. Drop the units and the number is almost meaningless.

Worked example

A car's position is \(s(t) = t^2\) meters, t in seconds. Using the difference-quotient limit from last lesson, \(s'(t) = 2t\). At \(t = 3\) s the velocity is \(s'(3) = \frac{ds}{dt}\big|_{t=3} = 2 \cdot 3 =\) 6 meters per second. The Leibniz form \(\frac{ds}{dt} = 2t\) says the same thing and displays the units directly: meters over seconds.

Your turn

Same car, later in the run: \(s(t) = t^2\) meters, and you want the velocity at \(t = 5\) s. The derivative is \(\frac{ds}{dt} = 2t\), so \(\left.\frac{ds}{dt}\right|_{t=5} = \) ____, in units of ____.

Reveal the answer

\(\left.\frac{ds}{dt}\right|_{t=5} = 2 \cdot 5 = \) 10 meters per second — units of s (meters) per unit of t (seconds). Drag the slider above to t = 5 and check it against the slope readout.

More info — why \(\frac{dy}{dx}\) isn't a fraction (but almost acts like one)

Last lesson defined the derivative as \(\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}\). Leibniz's symbol \(\frac{dy}{dx}\) deliberately looks like that ratio to remind you where it came from — and that memory is why the units work out to "y-units per x-unit," and why the notation feels natural in rate problems and (later) the chain rule. But no actual division happens: \(dy\) and \(dx\) aren't numbers you can split apart arbitrarily. Treat \(\frac{dy}{dx}\) as one symbol whose meaning is the difference-quotient limit. The OpenStax section in Dive deeper below compares all the notations side by side.

Check your understanding

Question 1 of 4

A ball's position is \(s(t) = t^2\) meters (t in seconds). What is its velocity at \(t = 4\) s?

Question 2 of 4

If \(C(q)\) is the cost in dollars to make \(q\) items, what does \(C'(q)\) represent, and in what units?

Question 3 of 4

Which statement about the notations \(f'(x)\), \(y'\), \(\frac{dy}{dx}\), and \(\frac{d}{dx}[f(x)]\) is correct?

Question 4 of 4

In Leibniz notation, \(\frac{dy}{dx}\) stands for exactly which quantity?

Recap

  • \(f'(x)\), \(y'\), \(\frac{dy}{dx}\), and \(\frac{d}{dx}[f]\) are four names for the same derivative — the difference-quotient limit from last lesson.
  • \(\frac{dy}{dx}\) is a single symbol, not literal division; to evaluate at a point, write \(\left.\frac{dy}{dx}\right|_{x=a}\).
  • In applications the derivative is a rate of change with units: the units of \(f\) per unit of \(x\).
  • Velocity is the derivative of position (meters per second); marginal cost is the derivative of cost (dollars per item). Always state the units.

Dive deeper

Sources

  • Derivative Notation and Interpretation as a Rate