Math Visualizations

What is a Derivative?

The derivative of a function measures how the function’s output changes as its input changes. It represents the rate of change or slope at any given point.

If $s(t)$ describes position over time, then the derivative $s'(t)$ gives the speed at time $t$:

$$s'(t) = \lim_{h \to 0} \frac{s(t+h) - s(t)}{h}$$

Position and Speed

In this visualization, the position function is $s(t) = t^2$. Its derivative is:

$$s'(t) = 2t$$

• The position graph (top) shows how distance accumulates over time
• The speed graph (bottom) shows the instantaneous rate of change
• The tangent line on the position graph has slope equal to the speed at that point

The Tangent Line

The tangent line at a point touches the curve at exactly that point and has the same slope as the curve. Its slope equals the derivative value:

$$\text{slope of tangent at } t = s'(t) = 2t$$

Try This

1. Drag the point on the position graph to change the time
2. Toggle the tangent line to see how its slope matches the speed value
3. Watch the speed graph — the dot shows the instantaneous speed at the current time
4. Notice that as time increases, the tangent gets steeper and the speed increases

Real-Life Applications

Derivatives appear everywhere in real life:

• Marginal cost in economics — the rate of change of cost with respect to quantity
• Velocity in physics — the derivative of position with respect to time
• Drug concentration rates in medicine — how fast a drug is metabolized
• Temperature change rates in engineering — how quickly a system heats or cools