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Unformatted text preview: 14.3: PARTIAL DERIVATIVES KIAM HEONG KWA Let f be a function of two variables whose domain D includes points arbitrarily close to ( a,b ). The partial derivative of f with respect to x is given by f x ( a,b ) = lim h f ( a + h,b )- f ( a,b ) h provided the limit exists. Note that we keep the variable y fixed in the limiting process. Likewise, the partial derivative of f with respect to y is given by f y ( a,b ) = lim h f ( a,b + h )- f ( a,b ) h provided the limit exists, where the variable x is kept fixed in taking the limit. The functions f x ( x,y ) = lim h f ( x + h,y )- f ( x,y ) h and f y ( x,y ) = lim h f ( x,y + h )- f ( x,y ) h are referred to as the partial derivatives of f . Remark 1. Note that by fixing the variable y , say y = b , one identifies the plane y = b from a multitude of planes parallel to the xz-plane. The intersection of this plane with the graph of f , i.e., the surface defined by z = f ( x,y ) , defines a curve that passes through the point ( a,b ) and is given by the relation z = f ( x,b ) (at least locally). In vector notation, this curve is given by r ( x ) = x i...
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