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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
holds. By the elimination method, we can ﬁnd ( ∂w )y and ( ∂w )x if w is not given
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:
= y z + xy e
= xz + y x2 exy .
∂y x 1 ...
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