l_notes11 - Lecture XI Chain Rule: Elimination Method Let w...

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Unformatted text preview: Lecture XI Chain Rule: Elimination Method Let w = f (x, y ) be a differentiable function of x and y . The linear approximation of f is given by Δfapp = fx (x, y )Δx + fy (x, y )Δy. We introduce a new notation, the differential notation for the increments Δf, Δx, Δy , namely we write df, dx, dy instead: df = fx dx + fy dy. This expression is called the differential of f. For example, the differential of w = x2 + y 2 − 1 is dw = 2xdx + 2ydy. For any function w = f (x, y ), the equality � � � � ∂w ∂w dw = dx + dy ∂x y ∂y x holds. By the elimination method, we can find ( ∂w )y and ( ∂w )x if w is not given ∂x ∂y directly as a function of x and y but can be reduced to such a function. We ilustrate this in the following example. Consider the these two equalities: z = g (x, y ) = exy . w = f (x, y, z ) = xyz, Then the differentials of w and z are dw = y zdx + xzdy + xydz and dz = y exy dx + xexy dy. Substituting dz in the first equality, we get dw = (yz + xy 2 exy )dx + (xz + y x2 exy )dy. Then the derivative of w with respect to x when w is seen as a function of x and y is precisely th term of dx in the equality above: � � � � ∂w ∂w 2 xy = y z + xy e and = xz + y x2 exy . ∂x y ∂y x 1 ...
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