Calculus
Partial Derivatives
A surface has a slope for every direction you could walk. Freeze y and slide a plane through it to get ∂f/∂x, freeze x for ∂f/∂y, then turn the plane to any direction u and watch D_u f = ∇f·u peak along the gradient.
y held still: ∂f/∂xx held still: ∂f/∂ydirection u: Dufgradient ∇f
Drag the point to move it · drag elsewhere to orbit · scroll to zoom · click a slice to select it
∂f/∂x at your point—
—
Surface
Which slice
Values
∂f/∂x—
∂f/∂y—
Duf = ∇f·u—
steepest slope |∇f|—
height f(x, y)—
What to observe
- Each plane freezes one variable, so inside it the surface is an ordinary curve (redrawn in the card). Its slope is the partial derivative.
- On Twist (f = 0.8xy), ∂f/∂x = 0.8y: slide y across zero and the slope along x flips sign.
- Pick Direction u and line it up with the green ∇f arrow: Duf = ∇f·u reaches its largest value, exactly |∇f|.
- Turn u a quarter turn, along the level curve through the point (the bold contour through it, on the surface and on the floor): the slice is flat there and Duf = 0.
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