Partial Derivatives Summary

Calculus 3 · Axiom Academy

Partial derivatives measure change one coordinate at a time; the gradient combines them to point toward steepest increase. Partials isolate effects: each partial measures sensitivity to one variable while every other variable is held fixed. The gradient unifies them: points toward maximum increase, and is that maximum rate. The chain rule handles composition: it extends single-variable calculus to functions built from other functions of several variables. Tangent planes approximate locally: the plane z=f(a,b)+f_x(x-a)+f_y(y-b) is the best linear approximation, generalizing the tangent line. Directional derivatives generalize slope: gives the rate of change in any direction, not just along the axes. Core Concept Functions of Several Variables A function like f(x,y) or f(x,y,z) depends on two or more independent inputs. For two variables its graph is a surface in 3D; for three we picture level surfaces. Level sets: f=c traces contours (2D) or level surfaces (3D) where the value is constant. Domain: the input points where f is defined — usually limited by square roots, denominators, or logarithms. Core Concept Limits & Continuity A 2D limit must reach the same value along every path to (a,b) , not just along the axes or a single line. Continuity means the limit exists and equals f(a,b) . Disprove a limit: find two paths ( y=mx , y=x^2 , the axes) that give different values. Polar trick: at the origin, set and let . Core Concept Partial Derivatives

This is the written version of the interactive lesson above. See the full Calculus 3 course.