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Unformatted text preview: Week 8 Partial Derivatives Definitions The partial derivatives of f ( x,y ) are defined as follows: f x : the partial derivative of f ( x,y ) with respect to x is the derivative of f ( x,y ) with y considered constant; f y : the partial derivative of f ( x,y ) with respect to y is the derivative of f ( x,y ) with x considered constant. The symbol is just pronounced d, and it signifies that were not dealing with a normal 1-variable derivative, but with a partial derivative. Some other notation for partial derivatives that we will use: f x = x f ( x,y ) = f x = f x ( x,y ) , f y = y f ( x,y ) = f y = f y ( x,y ) . Examples f ( x,y ) = x 2 + y f x = 2 x. After a bit of practice this will come naturally, but at first you can think of really putting in a constant c for y to see what the derivative looks like: f x = d dx ( x 2 + c ) = 2 x + 0 = 2 x . f ( x,y ) = xy f x = y ; here you can think of f x = d dx ( cx ) = c . Of course, in the real answer you have to make sure to put a y and not a c . All the usual rules for differentiation hold, like the Sum Rule, Product Rule and Quotient Rule: f ( x,y ) = xy 2 + x 3- 5 y f y = 2 xy- 5 , f ( x,y ) = ( x 2- y ) sin( x ) f x = 2 x sin( x ) + ( x 2- y ) cos( x ) , f ( x,y ) = y x + 3 y f y = 1 ( x + 3 y )- y 3 ( x + 3 y ) 2 = x ( x + 3 y...
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