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Partial Derivative

Undergraduate

Definition

Differentiating a multivariable function with respect to one variable while holding others constant.

(∂ f)/(∂ x) = lim_h → 0 (f(x+h, y) - f(x, y))/(h)

Definition of partial derivative w.r.t. x

∇ f = ( (∂ f)/(∂ x), (∂ f)/(∂ y) )

Gradient vector

Example 1

Find ∂f/∂x for f(x, y) = x²y + xy³.

Example 2

Find ∂f/∂y for f(x, y) = x²y + xy³.

Extrema of multivariable functions

Loss function minimization

Heat equation, wave equation