Transcription of 3.2 Higher Order Partial Derivatives
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3.2 Higher Order Partial Derivatives
Higher Order Partial DerivativesIffis a function of several variables, then we can find Higher Order partialsin the following (x,y)is a function of two variables, then f xand f yarealso functions of two variables and their partials can be taken. Hence we candifferentiate them with respect toxandyagain and find, 2f x2, the derivative offtaken twice with respect tox, 2f x y, the derivative offwith respect toyand then with respect tox, 2f y x, the derivative offwith respect toxand then with respect toy, 2f y2, the derivative offtaken twice with respect can carry on and find 3f x y2, which is taking the derivative offfirst withrespect toytwice, and then differentiating with respect tox, etc. In thismanner we can findnth- Order Partial Derivatives of a 2f x yand 2f y xare called mixed Partial Derivatives . They are equalwhen 2f x yand 2f y xare this course all the fuunctions we will encounter will have equalmixed Partial Find all partials up to the second Order of the functionf(x,y) =x4y2 : f x= 4x3y2 2xy6 f y= 2x4y 6x2y5 2f x2= 12x2y2 2y6 2f y x= 8x3y 12xy5 2f y2= 2x4 30x2y4 2f x y= 8x3 12xy5 Notations:fx= f xfy= f yfxx= 2f x2fyy= 2f y2fxy= 2f x Chain RuleYou are familiar with the chain rule for functions of one
3.2 Higher Order Partial Derivatives If f is a function of several variables, then we can find higher order partials in the following manner. Definition. If f(x,y) is a function of two variables, then ∂f ... Here is a link to the chapter on Higher Order Partial Differentiation.
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