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Unformatted text preview: Partial Differentiation John E. Gilbert, Heather Van Ligten, and Benni Goetz Single-variable case: for a function y = f ( x ) the derivative was defined by the limit f ( a ) = df dx x = a = lim h f ( a + h )- f ( a ) h of the Newtonian Quotient. The value of f ( a ) gave the rate of change of f ( x ) at x = a ; graphically, it was interpreted as the limit of the slope of secant lines passing through the point P ( a, f ( a )) as shown in green to the right below in the case of a parabola. Via the Point Slope formula the tangent line at P shown in orange became y = f ( a ) + f ( a )( x- a ) , and this provided a Linearization , L ( x ) = f ( a ) + f ( a )( x- a ) , of f that was useful in various estimates. In addition, first and second order derivatives turned out to be very helpful with determining graphs and with optimization. f ( a + h )- f ( a ) h P Multi-variable case: to differentiate a function z = f ( x, y ) of two variables or more we slice and use vectors to reduce matters to one variable. Lets do it first algbebraically: The First Order Partial Derivatives of z = f ( x, y ) at ( a, b ) are defined by f x ( a, b ) = f x ( a,b ) = lim h f ( a + h, b )- f ( a, b ) h , f y ( a, b ) = f y ( a,b ) = lim k f ( a, b + k )- f ( a, b ) k . In other words, we differentiate with respect to one variable exactly as in the one variable case, holding...
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