Section 7.2: Partial Derivatives

Partial Derivatives

• The partial derivative of f with respect to x, denoted ∂z/∂x, f_x, or f_x(x, y), is defined by:

  ∂z/∂x = limh → 0 [f(x + h, y) - f(x, y)]/h

• The partial derivative of f with respect to y, denoted ∂z/∂y, f_y, or f_y(x, y), is defined by:

  ∂z/∂y = limk → 0 [f(x, y + k) - f(x, y)]/h

Second-Order Partial Derivatives

If z = f(x, y), then:

fxx	= fxx(x, y)	= ∂2z/∂x2	= ∂/∂x(∂z/∂x)
fxy	= fxy(x, y)	= ∂2z/∂y∂x	= ∂/∂y(∂z/∂x)
fyx	= fyx(x, y)	= ∂2z/∂x∂y	= ∂/∂x(∂z/∂y)
fyy	= fyy(x, y)	= ∂2z/∂y2	= ∂/∂y(∂z/∂y)