Partial Derivatives

Partial derivatives and gradient of f(x,y)

Use x and y with + − * / ^ and functions like sin, cos, tan, ln, log, sqrt, exp. Example: x^2 + y^2

f(x, y) = x^2 + y^2 ∇f = (2*x, 2*y)
Partial derivatives
∂f/∂x =2*x= 2 at (1, 2)
∂f/∂y =2*y= 4 at (1, 2)
Gradient ∇f(x₀, y₀)
(2, 4)
f(x₀, y₀)
5
Magnitude |∇f|
4,472135955
Steepest ascent
63,4349488229° from the +x axis
Second-order partials
∂²f/∂x²
2
∂²f/∂y²
2
∂²f/∂x∂y (mixed)
0
Heat map of f(x, y) with the gradient arrow at (x₀, y₀)Heat map of f shaded by value, brighter where f is larger, with an arrow at the chosen point pointing in the direction of steepest ascent, perpendicular to the level curve. f(x, y) = x^2 + y^2.xy∇f(1, 2)