Partial Derivative: Interactive 3D Visualization

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Partial derivative in this visualization

For a function of two variables $f(x, y)$ , the partial derivative with respect to x at $(a, b)$ , written $\partial f/\partial x$ or $f_x$ , is the ordinary derivative of the one-variable function $x \mapsto f(x, b)$ at $x = a$ : y is held fixed and only x changes. The partial derivative with respect to y holds x fixed in the same way. This page shows both on six surfaces.

Partial derivative

\frac{\partial f}{\partial x}(a, b) = \lim_{h \to 0} \frac{f(a + h,\, b) - f(a, b)}{h} = \frac{d}{dx} f(x, b)\Big|_{x = a}

Also written $f_x(a, b)$ (or $f'_x(a, b)$ ), and the point $(a, b)$ as $(x_0, y_0)$ . The glass surface is the graph of $z = f(x, y)$ , with its contour map on the floor. The vertical plane $y = b$ through the blue point P cuts the surface along a curve, the slice: the graph of $x \mapsto f(x, b)$ . The grey arrow is a run of 1 in the x-direction; the rise from its tip to the blue tangent line, a signed vertical change (up positive, down negative), is $\partial f/\partial x$ , the slope of the slice at P. The window on the view draws the same slice as an ordinary graph. Choosing $\partial f/\partial y$ turns the plane to $x = a$ .

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Things to notice

A partial derivative can depend on both coordinates of P, including the one held fixed. Move the slice and the curve changes, and its slope can change with it: on $f = xy/4$ every slice along x is a straight line, and its slope $\partial f/\partial x = y/4$ depends only on which slice you take.
$\partial f/\partial x = 0$ says only that the slice along x is flat at P. Where a differentiable function has a local maximum or minimum at an interior point of its domain, both partial derivatives are 0; but that is not enough on its own: at a saddle point both are 0 too.
The rise is measured on the tangent line, not on the surface: over a step $h$ the slice itself changes by about $h\,\partial f/\partial x$ , closer the smaller $h$ is.
The 3D view keeps true proportions, z on the same scale as x and y, but how steep a line looks on screen depends on the angle you view it from. The slice window draws the slice face-on with equal scales, so slopes can be compared there directly.

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Where it is used

Together the two partial derivatives make the gradient $\nabla f = (f_x, f_y)$ . Where f is differentiable, the two tangent lines shown with the lever on Both lie in the tangent plane $z = f(a, b) + f_x\,(x - a) + f_y\,(y - b)$ . Having both partial derivatives is not enough on its own for a tangent plane to exist.
Whenever a quantity depends on several others, a partial derivative measures its response to one of them with the rest held fixed, such as how fast a gas’s pressure rises with temperature at constant volume.

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Controls

Drag P on the surface or its foot on the map. Key points jump to exact values; at a key critical point both partial derivatives are exactly 0.
The lever in Slice chooses which variable changes: $\partial f/\partial x$ holds y and slices along x, $\partial f/\partial y$ holds x and slices along y, and the middle, Both, shows the two slices together. Slice graph opens or closes the window; with one slice, P can also be dragged along it there. Of the two wheels in Point, one walks P along the slice and the other moves the cut. Presentation mode opens a full-screen view for projectors (← → walk P along the slice, ↑ ↓ move the cut, X, Y and B move the lever, H hides the answers).

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Partial derivative in this visualization

Things to notice

Where it is used

Controls

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