# 14.3 Notes - Section 14.3 Partial Derivatives Shubhodeep...

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Section 14.3 Partial Derivatives Shubhodeep Mukherji We define the partial derivatives of f wrt x at the point ( x 0 , y 0 ) as the ordinary derivative of f ( x, y 0 ) wrt x at the point x = x 0 . Use as the symbol. The Partial Derivative of f ( x, y ) wrt x at the point ( x 0 , y 0 ) is ∂f ∂x | ( x 0 ,y 0 ) = lim h 0 f ( x 0 + h, y 0 ) - f ( x 0 , y 0 ) h , provided the limit exists. The slope of the curve z = f ( x, y 0 ) at the point P ( x 0 , y 0 , f ( x 0 , y 0 )) in the plane y = y 0 is the value of the partial derivative of f wrt x at ( x 0 , y 0 ). The tangent line to the curve at P is the line in the plane y = y 0 that passes through P with this slope. The partial derivative ∂f ∂x at ( x 0 , y 0 ) gives the rate of change f wrt x when y is held fixed at the value y 0 . The Partial Derivative of f ( x, y ) wrt y at the point ( x 0 , y 0 ) is ∂f ∂y | ( x 0 ,y 0 ) = d dy f ( x 0 , y ) | y = y 0 = lim h 0 f ( x 0 , y 0 + h ) - f ( x 0 , y 0 ) 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. At the point (4 , - 5) is 2(4) + 3( - 5) = - 7 .

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