1Physiology 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) • 00//1),(yyxxkxyxr++= where k and y0 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) • txettxc4/2/12),(−−= • at small times, the solute is concentrated near x = 0, then it spreads out2Partial derivatives • for a function of a single variable, the derivative is xxfxxfxdxxdfΔ−Δ+→Δ=)()(0lim)( • for a function of two variables xyxfyxxfxxyxfΔ−Δ+→Δ=∂∂ ),(),(0lim),( • y is held constant and the partial derivative is with respect to x only • can also take multiple derivatives: ⎟⎠⎞⎜⎝⎛∂∂∂∂=⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂=∂∂∂⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂=∂∂⎟⎠⎞⎜⎝⎛∂∂∂∂=∂∂xfyyfxyxfyfyyfxfxxf22222 • usual rules for taking derivatives apply (sums, products, quotients, chain rule) • example: if2/12/12/121,)(),(−=∂∂= xyxzxyyxz (y is treated as a constant) • example: if bbbaebeaabababaeaabba 62)32(,3),(2222+=+∂∂=⎟⎠⎞⎜⎝⎛∂∂∂∂=∂∂∂+=φφφ • example: show that txettxc4/2/12),(−−= satisfies the equation 22xctc∂∂=∂∂ )4/(21224/2/14/2/322txetettctxtx −−−−+−=∂∂ )2/(4/2/12txetxctx−=∂∂−−
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