3 Rates of Change and Derivatives

From Change over an Interval to a Rate

Rates of change compare how much an output changes with how much its input changes. If $f$ is a function and the input changes from $a$ to $b$, the average rate of change is

The numerator is the change in the dependent variable, and the denominator is the change in the independent variable. Geometrically, this quotient is the slope of the secant line through $(a,f(a))$ and $(b,f(b))$.

For example, suppose an object's position is

where position is measured in meters and time in seconds. From $t=0$ to $t=2$, the average velocity is

This means the object's overall change in position was $5\text{ m}$ per second during the interval. It does not mean that the object necessarily moved at exactly $5\text{ m/s}$ at every moment.

Takeaway: An average rate describes change across an interval and is represented by a secant slope.

Instantaneous Change and the Derivative

The instantaneous rate of change describes how a quantity changes at one particular input value. To estimate it at $x=a$, use average rates over increasingly short intervals:

where $h\neq 0$. As $h$ approaches zero, the secant lines approach the tangent line. If the limiting value exists, the derivative of $f$ at $a$ is

The derivative function gives this rate at every input where it exists:

The notation $h\to 0$ does not mean substituting $h=0$ into the quotient. At $h=0$, the quotient would involve division by zero. Instead, the limit examines values for nonzero $h$ that become arbitrarily close to zero.

A function is differentiable at a point when its derivative exists there. Sharp corners, cusps, discontinuities, and certain vertical tangents can prevent differentiability.

Takeaway: The derivative is the limiting version of an average rate and represents the slope of the tangent line.

Finding Derivatives

The limit definition can be used directly to find a derivative. Let

Then

Thus, the instantaneous rate of change of $x^2$ at $x=a$ is $2a$. At $x=3$,

The graph therefore has tangent-line slope $6$ at $x=3$.

For routine calculations, derivative rules are more efficient. The constant rule gives

and the power rule gives

The sum and difference rules allow differentiation term by term. For example, if

then

Takeaway: Use the limit definition to understand the derivative and standard rules to calculate it efficiently.

Units and Interpretation

The units of a derivative are the output units divided by the input units:

For example, position in meters divided by time in seconds gives meters per second. Cost in dollars divided by number of items gives dollars per item. A derivative of $12$ dollars per item means that near the specified production level, one additional item is predicted to increase cost by approximately $12$ dollars, assuming the model is appropriate.

The sign and magnitude also matter. A positive derivative indicates that the output is increasing locally; a negative derivative indicates that it is decreasing locally. The magnitude indicates how rapidly the change occurs.

When interpreting a derivative over a small finite change, use language such as “approximately.” A derivative describes local behavior, so it predicts what happens near the specified input rather than guaranteeing the exact change over a large interval.

For example, if a population function satisfies

where population is measured in people and time in years, then at year $5$ the population is increasing at approximately $240$ people per year. If

in degrees Celsius per hour, then at hour $10$ the temperature is decreasing at approximately $1.5$ degrees Celsius per hour.

Takeaway: A meaningful derivative interpretation states the direction, numerical rate, units, and local context.

Derivatives in Motion

For motion, let $s(t)$ represent position as a function of time. Then velocity is the first derivative:

and acceleration is the second derivative:

The signs give qualitative information:

• If $v(t)>0$, position is increasing and the object moves in the positive direction.

• If $v(t)<0$, position is decreasing and the object moves in the negative direction.

• If $a(t)>0$, velocity is increasing.

• If $a(t)<0$, velocity is decreasing.

An object is speeding up when velocity and acceleration have the same sign. It is slowing down when they have opposite signs.

For

we obtain

and

At $t=2$ seconds,

The object is moving in the positive direction, and its velocity is increasing. Since velocity and acceleration are both positive at this time, the object is speeding up.

Takeaway: Position, velocity, and acceleration form a chain of successive derivatives, and their signs must be interpreted together.

Local Approximation and Problem Solving

A derivative can be used to estimate a nearby function value through a linear approximation. If $f(a)$ and $f'(a)$ are known, then for a small change $h$,

This is the equation of the tangent-line model near $x=a$.

Suppose a machine has produced $f(100)=250$ units after $100$ minutes and has production rate $f'(100)=4$ units per minute. After approximately $3$ more minutes,

The estimate is most reliable when the change in input is small and the production rate does not change too dramatically during those three minutes.

For rate-of-change problems, use this sequence:

1. Define what the input and output represent.

2. Decide whether the question concerns an interval or a single point.

3. Choose an average-rate quotient or a derivative accordingly.

4. Attach the correct units.

5. Interpret the sign and magnitude in context.

6. Consider whether the model is reliable over the interval or near the point being studied.

Takeaway: Derivatives provide both exact local rates and practical predictions, but local approximations should not be extended carelessly over large changes.