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Derivatives of Multi-Variable Functions: Step-by-Step Analysis

In this tutorial, we explore the first and second derivatives of two specific functions of two variables. The first function is f(x,y) = 7 + 5x + 2y – 5x² – 4y² – 11xy, and we derive its first derivatives fx and fy, as well as second derivatives fxx, fyy, and fxy. The second function f(x,y) = –5 – 2x + y – x² – 3y² + 2xy is analyzed in a similar manner. This detailed breakdown will help you understand the process of finding derivatives for functions of multiple variables.

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Derivatives of Multi-Variable Functions: Step-by-Step Analysis

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  1. Think Break #5 • Give the 1st and 2ndderivatives fx(x,y), fy(x,y),, fxx(x,y), fyy(x,y), fxy(x,y) of each function: • f(x,y) = 7 + 5x + 2y – 5x2 – 4y2 – 11xy • f(x,y) = – 5 – 2x + y – x2 – 3y2 + 2xy

  2. Think Break #5:Answer • f(x,y) = 7 + 5x + 2y – 5x2 – 4y2 – 11xy • fx(x,y) = 5 – 10x – 11y • fy(x,y) = 2 – 8y – 11x • fxx(x,y) = – 10, fyy(x,y) = – 8, fyy(x,y) = – 11 • f(x,y) = – 5 – 2x + y – x2 – 3y2 + 2xy • fx(x,y) = – 2 – 2x + 2y • fy(x,y) = 1 – 6y + 2x • fxx(x,y) = – 2, fyy(x,y) = – 6, fyy(x,y) = 2

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