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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 vextendsingle vextendsingle vextendsingle 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. Let’s 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 vextendsingle vextendsingle vextendsingle ( a,b ) = lim h → f ( a + h, b ) − f ( a, b ) h , f y ( a, b ) = ∂f ∂y vextendsingle vextendsingle vextendsingle ( 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 the other variables fixed. After freeing the fixed variable the partial derivativesthe other variables fixed....
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- Spring '11
- Derivative, fxy, order partial derivatives, fyx