Independent & Dependent Variables · dy/dx · The Tap & Pipe Analogy
An independent variable is one you freely choose or control. It doesn't react to anything else. Think of the tap handle — you turn it wherever you like. In math: x. A tiny change in x is called dx.
A dependent variable is determined by x — it cannot decide its own value. Think of the water flowing from the pipe: open the tap more (↑x) → more water (↑y). In math: y = f(x). A tiny change in y is called dy.
Drag the slider to open or close the tap (x). The water stream and pool react instantly — that's y = x² responding to x.
Click a rule — see how it connects to the tap analogy.
Bring the exponent down as a multiplier, then reduce the power by 1. This covers y = x², y = x³, y = x⁵ and beyond. Opening the tap wider causes increasingly rapid flow increases.
A straight-line tap: every equal nudge to x gives exactly m more units of flow — always the same, no acceleration. The constant b (starting offset) vanishes because d/dx(constant) = 0. The tangent line IS the curve itself.
A rapidly accelerating tap. At x = 2 the rate is 12, at x = 5 the rate is 75. The derivative 3x² is itself a function of x — the rate of change keeps changing. Opening a nearly-full tap even slightly causes a huge surge in flow.
The most magical function in calculus: its derivative equals itself! The flow rate equals the current flow. The more water that flows, the faster it grows. This self-referential property makes eˣ appear in compound growth, population models, and physics everywhere.
A diminishing-returns tap: fast at first, then slowing down. Opening from x = 0→1 adds 1 unit of flow, but x = 9→10 adds only 0.16. This is the Power Rule with n = ½: dy/dx = ½ · x−½ = 1/(2√x). The derivative shrinks as x grows.
A constant doesn't react to x at all — the tap is welded shut. y stays the same no matter what x is. Rate of change = zero. On a graph: a perfectly flat horizontal line with slope 0. Shifting a function up or down (adding a constant) never changes its derivative.
Choose a function below. Orange dot = your current (x, y). Green dashed = tangent — its slope IS dy/dx.
When one quantity goes up, the other goes down — but their product stays constant. Drag the slider and watch the geometry.
The curve never touches the axes — as x→∞, y→0 and as x→0, y→∞. The shaded rectangle always has the same area = k.
Click the correct answer for each question.
1. In the tap analogy, what is the independent variable?
2. If y = x², what is dy/dx?
3. If y = x³, what is dy/dx?
4. If y = 2x + 1, what is dy/dx?
5. What is special about the derivative of y = eˣ?
6. If y = √x, what is dy/dx at x = 4?
7. dy/dx = 0 at a point means the function is…