Partial Derivatives
Differentiate with respect to one variable while holding the others fixed. Mixed second partials agree for the functions met in practice, which is a statement worth noticing rather than assuming.
What a learner can do afterwards
- Compute both first partials of a function of two variables
- Interpret a partial derivative as a slope along a coordinate direction
- Check that the two mixed second partials agree for a given function
1 · Read
A partial derivative measures slope in one coordinate direction while the other variable stays frozen. To get fx, treat every y as a constant and differentiate in x. For x squared + 3xy + y squared, fx = 2x + 3y, so fx(2, 1) = 7.
To get fy, freeze x instead. For the same f, fy = 3x + 2y, so fy(2, 1) = 8. Try g = 5x squared + 4xy minus 7y cubed: gy = 4x minus 21y squared, so gy(1, 1) = minus 17.
Geometrically, fx(a, b) is the slope of the trace you cut by holding y fixed at b. That trace is an ordinary single variable curve. Ana is right to picture fx as its slope.
Second partials differentiate twice, and for smooth functions the mixed orders agree. For x squared y cubed, fx = 2x y cubed and fxy = 6x y squared gives 24 at (1, 2). If your fxy and fyx ever differ, recompute the first partials.
Freeze one variable, differentiate in the other, and let matching mixed partials confirm your algebra.
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.