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105-8a-PartialDerivatives

105-8a-PartialDerivatives - Week 8 Partial Derivatives...

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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 we’re 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 ) 2 .
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