105-8a-PartialDerivatives

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

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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 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...
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This note was uploaded on 01/14/2012 for the course MATH 105 taught by Professor Malabikapramanik during the Fall '10 term at UBC.

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

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