Partial derivatives: ∂f/∂x treats y as constant, differentiates w.r.t. x. Gradient: ∇f = (∂f/∂x, ∂f/∂y). Critical points where ∇f = 0. Second partials test: D = fₓₓfᵧᵧ − fₓᵧ² determines nature of critical point.
How to Use
Select a function:
- f(x,y): Choose preset
- Point: (x₀, y₀)
- Output: ∂f/∂x, ∂f/∂y, ∇f
Differentiation Rules
- ∂/∂x[xⁿ] = nxⁿ⁻¹
- ∂/∂x[xy] = y
- ∂/∂x[sin(xy)] = y·cos(xy)
- Chain rule applies normally
The Gradient
∇f points in the direction of steepest ascent. |∇f| gives the rate of maximum increase. Perpendicular to level curves. Central to machine learning (gradient descent).
Step-by-Step Instructions
- 1Select f(x,y).
- 2Enter point (x₀,y₀).
- 3View ∂f/∂x.
- 4View ∂f/∂y.
- 5See gradient vector.