Derivative as Rate of Change

The Derivative as Rate of Change

Derivatives measure how fast things change. This is their most powerful real-world application!

Average vs. Instantaneous Rate of Change

Average rate of change (over an interval): $\frac{f(b) - f(a)}{b - a}$

Instantaneous rate of change (at a point): $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

The derivative is the instantaneous rate of change!

Motion: Position, Velocity, Acceleration

This is where derivatives truly shine!

If $s(t)$ is position (where you are):

• $s'(t) = v(t)$ is velocity (how fast you're moving)
• $v'(t) = s''(t) = a(t)$ is acceleration (how fast your speed is changing)

The derivative hierarchy: $\text{Position} \xrightarrow{\text{derivative}} \text{Velocity} \xrightarrow{\text{derivative}} \text{Acceleration}$

Example: Free Fall

An object is dropped from a building. Its height is $h(t) = 100 - 4.9t^2$ meters (where $t$ is in seconds).

Find the velocity: $v(t) = h'(t) = -9.8t \text{ m/s}$

Find velocity at $t = 2$ seconds: $v(2) = -9.8(2) = -19.6 \text{ m/s}$

The negative sign means downward!

Find the acceleration: $a(t) = v'(t) = h''(t) = -9.8 \text{ m/s}^2$

This is constant (gravity)!

Sign of the Derivative

The sign tells you the direction of change:

| $f'(x)$ | Meaning | |---------|---------| | $f'(x) > 0$ | $f$ is increasing | | $f'(x) < 0$ | $f$ is decreasing | | $f'(x) = 0$ | $f$ is stationary (not changing) |

Example: If $v(t) > 0$, you're moving in the positive direction. If $v(t) < 0$, you're moving in the negative direction.

Units Matter!

The derivative's units are: $\frac{\text{units of output}}{\text{units of input}}$

Examples:

• Position in meters, time in seconds → Velocity in m/s
• Cost in dollars, quantity in items → Marginal cost in $/item
• Temperature in °C, time in hours → Rate of temperature change in °C/hr

Related Rates

When two quantities both depend on time, their rates of change are related through derivatives!

Example: A balloon is being inflated. The radius increases at 2 cm/s. How fast is the volume changing?

We know:

• $\frac{dr}{dt} = 2$ cm/s (given)
• $V = \frac{4}{3}\pi r^3$ (volume formula)

Take the derivative with respect to time: $\frac{dV}{dt} = 4\pi r^2 \cdot \frac{dr}{dt}$

At $r = 5$ cm: $\frac{dV}{dt} = 4\pi(5)^2(2) = 200\pi \text{ cm}^3\text{/s}$

Economics: Marginal Analysis

In economics, "marginal" means "derivative"!

Marginal cost: $C'(x)$ = cost to produce one more unit Marginal revenue: $R'(x)$ = revenue from selling one more unit Marginal profit: $P'(x) = R'(x) - C'(x)$

Example: If $C(x) = 1000 + 5x + 0.01x^2$ dollars: $C'(x) = 5 + 0.02x$

At $x = 100$ units: C'(100) = 5 + 2 = \7$ per unit

Population Growth

If $P(t)$ is population at time $t$:

• $P'(t) > 0$: Population growing
• $P'(t) < 0$: Population shrinking
• $|P'(t)|$: Rate of growth/decline

Example: $P(t) = 1000e^{0.02t}$ (exponential growth) $P'(t) = 20e^{0.02t}$

The population is always growing (positive derivative)!

Optimization Preview

To maximize or minimize something:

1. Find where $f'(x) = 0$ (critical points)
2. Test to see if it's a max or min
3. Check endpoints if on a closed interval

We'll explore this deeply in later lessons!

The Power of Derivatives

Derivatives let us:

• ✓ Predict future behavior
• ✓ Find optimal solutions
• ✓ Understand dynamic systems
• ✓ Model real-world phenomena
• ✓ Make informed decisions

Calculus transforms static snapshots into dynamic understanding!