{"id":4383,"date":"2013-12-18T06:05:33","date_gmt":"2013-12-17T21:05:33","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4383"},"modified":"2014-08-01T19:14:58","modified_gmt":"2014-08-01T10:14:58","slug":"partial-derivatives","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4383","title":{"rendered":"Partial derivatives"},"content":{"rendered":"
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The partial derivatives of f(x, y) with respect to x and y are defined by<\/p>\n

\\displaystyle \\frac{\\partial f}{\\partial x}= \\lim\\limits_{h \\rightarrow 0} \\frac{f(x + h, y) - f(x, y)}{h}\\\\\\vspace{0.2 in} \\frac{\\partial f}{\\partial y} = \\lim\\limits_{k \\rightarrow 0} \\frac{f(x, y + k) - f(x, y)}{k}<\/p>\n

if these limits exist. It’s often written h = Δx, k = Δy. Note that \\partial f\/\\partial x is simply the ordinary derivative of f with respect to x keeping y constant, while \\partial f\/\\partial y is the ordinary derivative of f with respect to y keeping x constant. <\/p>\n

Higher derivatives are defined similarly. For example, you have the second order derivatives<\/p>\n

\\displaystyle \\frac{\\partial}{\\partial x}\\left(\\frac{\\partial f}{\\partial x}\\right) = \\frac{\\partial^2f}{\\partial x^2}\\\\\\vspace{0.2 in} \\frac{\\partial}{\\partial x}\\left(\\frac{\\partial f}{\\partial y}\\right) = \\frac{\\partial^2f}{\\partial x\\partial y}\\\\\\vspace{0.2 in} \\frac{\\partial}{\\partial y}\\left(\\frac{\\partial f}{\\partial x}\\right) = \\frac{\\partial^2f}{\\partial y\\partial x}\\\\\\vspace{0.2 in} \\frac{\\partial}{\\partial y}\\left(\\frac{\\partial f}{\\partial y}\\right) = \\frac{\\partial^2f}{\\partial y^2}<\/p>\n

The deviation are sometimes denoted fx<\/sub> and fy<\/sub>. In such case fx<\/sub>(a, b), fy<\/sub>(a, b) denote these partial derivatives evaluated at (a, b). <\/p>\n

The deviations are denoted by fxx<\/sub>, fxy<\/sub>, fyx<\/sub>, fyy<\/sub> respectively. The second and third results will be the same if f has continuous partial derivatives of second order at least. <\/p>\n

The differentiation of f(x, y) is defined as<\/p>\n

\\displaystyle df = \\frac{\\partial f}{\\partial x}dx + \\frac{\\partial f}{\\partial y}dy<\/p>\n

where h = Δx = dx, k = Δy = dy.<\/p>\n<\/blockquote>\n