Equation field guide
Derivatives: instantaneous rate of change
A derivative describes how quickly one quantity changes as another changes. It is the slope of a curve at a particular point.
From average to instantaneous change
The average rate of change from x to x + h is [f(x + h) − f(x)]/h. A derivative takes the limiting value as h becomes very small: f′(x) describes the local rate at x.
For position over time, the derivative is velocity. For a temperature curve, it is the rate of warming or cooling. Units therefore change: metres divided by seconds becomes metres per second.
Common notation
f′(x), df/dx, and d/dx f(x) all denote derivatives, with small differences in emphasis. ∂ is used for a partial derivative when a function has several input variables and the others are held fixed.
For example, d(x²)/dx = 2x. At x = 3, the slope is 6: a small positive change in x produces approximately six times that change in x².
What it does not say
A derivative is local. A positive derivative at one point says the function is increasing there; it does not prove the function increases everywhere.
A derivative can fail to exist at a sharp corner, jump, or vertical tangent. The formula’s assumptions determine whether derivative rules apply.