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Unformatted text preview: Section 14.3 Partial Derivatives Shubhodeep Mukherji We define the partial derivatives of f wrt x at the point ( x ,y ) as the ordinary derivative of f ( x,y ) wrt x at the point x = x . Use ∂ as the symbol. The Partial Derivative of f ( x,y ) wrt x at the point ( x ,y ) is ∂f ∂x | ( x ,y ) = lim h → f ( x + h,y )- f ( x ,y ) h , provided the limit exists. The slope of the curve z = f ( x,y ) at the point P ( x ,y ,f ( x ,y )) in the plane y = y is the value of the partial derivative of f wrt x at ( x ,y ). The tangent line to the curve at P is the line in the plane y = y that passes through P with this slope. The partial derivative ∂f ∂x at ( x ,y ) gives the rate of change f wrt x when y is held fixed at the value y . The Partial Derivative of f ( x,y ) wrt y at the point ( x ,y ) is ∂f ∂y | ( x ,y ) = d dy f ( x ,y ) | y = y = lim h → f ( x ,y + h )- f ( x ,y ) h , provided the limit exists. Example 1 Find the values of ∂f ∂x and ∂f ∂y at the point (4 ,- 5) for the function f ( x,y ) = x 2 + 3 xy + y- 1 . To find ∂f ∂x , we treat y as a constant and differentiable wrt x : ∂f ∂x = ∂ ∂x ( x 2 + 3 xy + y- 1) = 2 x + 3(1)( y ) + 0- 0 = 2 x + 3 y....
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This note was uploaded on 08/03/2011 for the course ECON 101 taught by Professor Profeessor during the Spring '11 term at Aachen University of Applied Sciences.
- Spring '11