What Are the Different Notations for a Derivative (dy/dx; f'(x))?

The notations $\frac{dy}{dx}$ and (f'(x)) both represent a derivative: the instantaneous rate of change of a function with respect to its independent variable. They express the same central idea but emphasize different features of the relationship. The derivative function assigns a gradient to each value in the function's domain where the derivative exists.

This is part of IB Mathematics: Analysis and Approaches, SL 5.1: Introduction of differential calculus, and can be assessed on either Paper 1 or Paper 2.

For example, if $f(x)=x^3$, then

$$f'(x)=\frac{dy}{dx}=3x^2.$$

At $x=2$, the derivative is

$$f'(2)=3(2)^2=12.$$

Graphically, this means that the tangent to $y=f(x)$ at $x=2$ has gradient $12$. Notice that (f'(x)) is a function, whereas $f'(2)$ is the numerical value of that derivative at one point.

In context, derivative units are “output units per input unit.” For example, if $s$ is measured in metres and $t$ in seconds, then $\frac{ds}{dt}$ is measured in metres per second.

A common misconception is that $\frac{dy}{dx}$ is an ordinary fraction. It is derivative notation, although its fraction-like structure becomes useful in methods such as the chain rule.

Exam technique: Use notation consistently and identify the variables correctly. When asked to interpret a derivative, state its meaning as a gradient or rate of change, including units where a context is given.