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Partial derivative and gradient

A partial derivative differentiates with respect to one variable at a time, holding the others constant. The gradient ∇f\nabla f collects all partial derivatives into a vector pointing in the direction where ff increases fastest.

fx=∂f∂xf_x = \frac{\partial f}{\partial x}partial derivative w.r.t. x (y held constant)
∇f=(fx,fy)\nabla f = (f_x, f_y)the gradient

Symbols

∇f\nabla fgradient vector, points towards steepest increase

Example

f=x2+y2f=x^2+y^2 at (3,4)(3,4): ∇f=(6,8)\nabla f=(6,8), and ∣∇f∣=36+64=10|\nabla f|=\sqrt{36+64}=10.

The gradient is always perpendicular to the level curves of ff.
Practise partial derivatives for free →

The second derivative test →

Part of Multivariable Calculus: Partial derivatives.