Notes-Partial Differentiation

Notes-Partial Differentiation - Review: Partial...

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Review: Partial Differentiation Suppose f is a function of two, or more, independent variables. At each point within its domain, the function could have different instantaneous rates of change, in different directions we trace. These directional derivatives could be computed using the instantaneous rates of change of f along the directions of the coordinate axes (of the independent variables): the rates of change along those “principal directions” are called the partial derivatives of f . For a function of two independent variables, f ( x , y ), the partial derivative of f with respect to x can be found by applying all the usual rules of differentiation. The only exception is that, whenever and wherever the second variable y appears, it is treated as a constant in every respect. The partial derivative of f with respect to y can similarly be found by treating x as a constant whenever it appears. For a function of more than two independent variables, the same method applies. Its partial derivative with respect to, say, the variable x , can be obtained by differentiating it with respect to x , using all the usual rules of differentiation. Except that all the other independent variables, whenever and wherever they occur in the expression of f , are treated as constants. Notations of partial derivatives: Partial derivative of f w.r.t. x x f f x Partial derivative of f w.r.t. y y f f y … is analogous to this familiar notation … x d f d f
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Example : Suppose f ( x , y ) = x 9 y 8 + 2 x + y 3 . Then f x = 9 x 8 y 8 + 2, f y = 8 x 9 y 7 + 3 y 2 . Example : Suppose f ( x , y ) = e xy - ln( xy ) + y 2 sin(4 x ) + 2 x 3 - 5 y . Then 2 2 6 ) 4 cos( 4 1 x x y x ye f xy x + + - = , 5 ) 4 sin( 2 1 - + - = x y y xe f xy y .
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Higher Order Partial Derivatives A partial derivative of f could be differentiated again with respect to any of
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Notes-Partial Differentiation - Review: Partial...

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