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How do you find #(df)/dy# and #(df)/dx# of #f(x,y)=(4x-2y)/(4x+2y)#, using the quotient rule?

Answer 1

Evelyn Agard

#f_x (x,y) = (4y)/((2x + y)^2)#

#f_y (x,y) = (-4x)/((2x+y)^2)#

#f(x, y) = (2x - y)/(2x + y) = u/v#

#(partial f)/(partial x) = f_x = frac{u_x v - u v_x}{v^2} = frac{2(2x+y) - (2x-y)*2}{(2x + y)^2}#

#(partial f)/(partial y) = f_y = frac{u_y v - u v_y}{v^2} = frac{-1(2x+y) - (2x-y)*1}{(2x + y)^2}#

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Answer 2

Ezekiel Alexander

To find ((df)/dy) and ((df)/dx) of (f(x,y)=(4x-2y)/(4x+2y)) using the quotient rule:

Compute (df/dx): [df/dx = \frac{d}{dx} \left( \frac{4x - 2y}{4x + 2y} \right) = \frac{(4)(4x + 2y) - (4x - 2y)(4)}{(4x + 2y)^2}]
Compute (df/dy): [df/dy = \frac{d}{dy} \left( \frac{4x - 2y}{4x + 2y} \right) = \frac{(2)(4x + 2y) - (4x - 2y)(2)}{(4x + 2y)^2}]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.