{"id":4480,"date":"2013-12-25T06:05:18","date_gmt":"2013-12-24T21:05:18","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4480"},"modified":"2014-08-01T19:05:51","modified_gmt":"2014-08-01T10:05:51","slug":"leibnitzs-rule-for-differentiating-an-integral","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4480","title":{"rendered":"Leibnitz’s rule for differentiating an integral"},"content":{"rendered":"
\n

Let<\/p>\n

\\displaystyle I(a) = \\int_{a}^{b}f(x, \\alpha)dx<\/p>\n

where f<\/span> is supposed continuous and differentiable. Then Leibnitz’s rule<\/span> states that if a<\/span> and b<\/span> are differentiable functions of α, <\/p>\n

\\displaystyle \\frac{dI}{d\\alpha} = \\int_{a}^{b}\\frac{\\partial f}{\\partial \\alpha}dx + f(b, \\alpha)\\frac{db}{d\\alpha} - f(a, \\alpha)\\frac{da}{d\\alpha}<\/p>\n<\/blockquote>\n