∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 3 Theory

Multivariable Functions, Level Curves & Partial Derivatives

Contour elevation slices, Clairot's Theorem, and tangent approximations

Rotate around a 3D surface z = f(x,y) and click anywhere on the surface to compute the instantaneous partial derivatives fx and fy along orthogonal slices.

1. Partial Derivatives as Orthogonal Slices

To compute ∂f/∂x, freeze y as a constant and take the standard 1D derivative along the x-direction. Geometrically, this intersects the surface with a vertical plane y = y₀.

∂∂x[x3y2]=3x2y2,∂∂y[x3y2]=2x3y\frac{\partial}{\partial x}[x^3 y^2] = 3x^2 y^2, \qquad \frac{\partial}{\partial y}[x^3 y^2] = 2x^3 y

Quick Reference & Cheat Sheet

Clairaut's Theorem (Equality of Mixed Partials)
fxy=fyxf_{xy} = f_{yx}
Mixed partial derivatives are independent of differentiation order when continuous.