# UA PSIO 472 - Functions of more than one variable (2 pages)

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## Functions of more than one variable

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Lecture Notes

- Pages:
- 2
- School:
- University of Arizona
- Course:
- Psio 472 - Quantitative Modeling of Biological Systems

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Physiology 472 572 2011 Quantitative modeling of biological systems Lecture 3 Functions of more than one variable partial derivatives Functions of more than one variable f f x y where f is a dependent variable and x y are independent variables occur in many areas of biology where dependence on multiple variables is typical can visualize as a surface whose height is the dependent variable example a reaction whose rate depends on the concentrations of a reactant x and of an inhibitor y r x y kx where k and y0 1 x x0 y y 0 are constants watch units if x y are measured in M i e mol l and r has units of M s then k has units s 1 and x0 and y0 have units M example concentration of a diffusing solute c depends on location x and time t c x t t 1 2 e x at small times the solute is concentrated 2 4t near x 0 then it spreads out 1 Partial derivatives lim f x x f x df x x 0 dx x for a function of a single variable the derivative is for a function of two variables y is held constant and the partial derivative is with respect to x only can also take multiple derivatives 2 f f 2 x x x lim f x x y f x y f x y x 0 x x 2 f f 2 y y y 2 f f f x y x y y x usual rules for taking derivatives apply sums products quotients chain rule example if z x y xy 1 2 example if a b ab 3a e 2 example show that c x t t z 1 1 2 1 2 y x y is treated as a constant x 2 2 b 1 2 e 2 2ab 3a 2 e b 2b 6ae b a b a b a x 2 4t c 2 c satisfies the equation t x 2 2 2 c 1 t 3 2 e x 4t t 1 2 e x 4t x 2 4t 2 t 2 2 c t 1 2 e x 4t x 2t x 2 2 2c c t 1 2 e x 4t 1 2t t 1 2 e x 4t x 2 4t 2 2 t x 2

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