Instantaneous Rate of Change

An instantaneous rate is the value approached by average rates over intervals shrinking toward one point.

From an interval to an instant

Average rate of change needs two distinct inputs. To estimate a rate at $x=a$, compare $a$ with nearby inputs $a+h$:

As $h$ becomes closer to zero from both sides, the secant slopes may approach one number. That limit is the instantaneous rate of change.

Limit definition

When the limit exists,

The notation $f'(a)$ is read “$f$ prime at $a$.” The fraction itself is undefined at $h=0$, but the limit asks what values approach—not what happens after substituting zero directly.

Tangent-line meaning

Geometrically, secant lines through two graph points approach a tangent line at $(a,f(a))$. The tangent slope is $f'(a)$.

A tangent is not always a line that touches only once. It is the line giving the graph's local direction and best linear approximation near the point.

Worked example from first principles

The tangent line at $(3,9)$ has slope $6$.

Numerical estimation

Compute average rates on increasingly small intervals around $a$, such as $[a,a+0.1]$, $[a,a+0.01]$, and corresponding left intervals. If both sides approach the same value, that value estimates the instantaneous rate.

Using both sides can reveal a corner, cusp, jump, or other point where one tangent slope does not exist.

Graphical estimation

Draw a tangent line that follows the curve's local direction, choose two convenient points on the tangent, and calculate rise over run. The selected points need not both lie on the original curve; they lie on the tangent line.

Graphical estimates depend on scale and drawing accuracy, so report appropriate precision.

Units and interpretation

An instantaneous rate has the same output-per-input units as average rate of change. If $s(t)$ measures metres and $t$ measures seconds, then $s'(4)$ is instantaneous velocity in metres per second at $t=4$.

The input $a$, output $f(a)$, and rate $f'(a)$ are three different quantities with different meanings and often different units.

Where the rate may not exist

An instantaneous rate may fail to exist at a discontinuity, sharp corner, cusp, vertical tangent, or point where left and right secant slopes approach different values.

For $f(x)=|x|$ at $x=0$, slopes from the left approach $-1$ and from the right approach $1$, so no single derivative exists there.

Local linear approximation

Near $x=a$, a differentiable function behaves approximately like

This tangent-line approximation turns a complicated curve into a useful local linear model. Its accuracy generally decreases farther from $a$.

Common mistakes

Substituting $h=0$ before simplifying. The quotient then divides by zero.

Using only a large interval. Instantaneous rate requires a limiting process.

Confusing $f(a)$ with $f'(a)$. One is an output; the other is a slope.

Assuming every visible point has a tangent. Check left and right behaviour.

Using curve points to measure a drawn tangent slope incorrectly. Measure the tangent line itself.

Quick self-check

• Is the difference quotient formed correctly?
• Did I simplify for $h\ne0$ before taking the limit?
• Do left and right rates approach the same value?
• Does the sign match the graph's local direction?
• Are the units output per input?
• Is the tangent interpretation local rather than interval-wide?

Related topics

• Average Rate of Change
• Limits
• Derivative from First Principles
• Tangent Lines