Example 2: Find all second-order partial derivatives of...

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S econd- O rder P artial D erivatives Previously we have taken the partial derivative of a function f ( x , y ). But those partial derivatives were themselves functions and so we can take their partial derivatives. The Second-Order Partial Derivatives of z = f ( x , y ) 2 2 ( ) x x z z xx f f x x x 2 ( ) y x yx z z f f x y x y   2 ( ) x y z z xy f f y x y x   2 2 ( ) y y yy z z f f y y y Example 1: Compute the second-order partial derivatives of f ( x , y ) = x 2 y + 5 x sin( y ). Solution: Notice that f x ( x , y ) = 2 xy + 5sin( y ) and f y ( x , y ) = x 2 + 5 x cos( y ). Thus, we have that f xx ( x , y ) = 2 y f xy ( x , y ) = 2 x + 5cos( y ) f yx ( x , y ) = 2 x + 5cos( y ) f yy ( x , y ) = –5 x sin( y ). Notice that in Example 1 above, we have that f xy ( x , y ) = f yx ( x , y ). Indeed, this is typically always the case. Thus, it does not matter if we take the partial derivative with respect to x first or with respect to y first. The Equality of Mixed Partial Derivatives If f xy ( x , y ) and f yx ( x , y ) are continuous at ( a , b ), an interior point of their domain, then ( , ) ( , ) xy yx f a b f a b . 1
We can visualize this statement in the following diagram. z = f x,y z x z y f x x,y f y x,y f xy x,y = f yx x,y f xx x,y f yy x,y f x x f x y f y f y y x Figure 1: Equality of Mixed Partial Derivatives Example 2: Find all second-order partial derivatives of f ( x , y ) = ln(3 x + 5 y ). 5

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