Defining Partial Derivatives

Calculus 3 · Axiom Academy

LESSON Defining Partial Derivatives A partial derivative is just an ordinary slope — you hold every variable but one still, and read the tilt of the curve that survives. 1. Freeze y , Then It's Just a Derivative Fix y at some value b and let only x move. Now f(x,b) is a plain single-variable function, so we can take its derivative the old way — a limit of slopes of secant lines as the step h shrinks to zero. That limit is the partial derivative with respect to x . Hold y constant; only x is nudged by h Symmetrically, hold x constant to define Here is what "freeze y " looks like on the surface. Cut z = f(x,y) with the vertical plane y = b . The surface and the plane meet along one curve — the x -trace — and the slope of that curve at the point (a,b) is . You are measuring how steep the hill is if you walk parallel to the x -axis. y = b is held fixed — a vertical wall parallel to the xz -plane. Where the wall meets the surface: z = f(x,b) , a curve in that plane. Its slope at x = a is the number . Positive surface rises in the x -direction; negative it falls. The animation uses . On the slice y=b the trace is , whose slope is 2x — so , matching the tilting tangent. Swap the roles. Now freeze x = a and cut with the plane x = a . The surface meets it along the y -trace z = f(a,y) , and its slope at (a,b) is — the steepness if you walk parallel to the y -axis instead. Same surface, same point, a different direction and generally a different slope.

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