Chapter 4: Partial Differentiation
Section 4.3: Chain Rule
If z=x y+x fy/x, show that x zx+y zy=x y+z.
It is most convenient to define wx,y=y/x so that z=x y+x fwx,y.
The following calculation then results from an application of the chain rule.
x zx+y zy
=x y+x f′wx+f+y x+x f′wy
=x y+x f′⋅−y/x2+f+y x+x f′⋅1/x
=x y−y f′+x f+x y+y f′
=2 x y+y−yf′+x f
=2 x y++x f+0⋅f
=2 x y+x f
=2 x y+z−x y
Maple Solution - Interactive
Define w and z
Context Panel: Assign Name
Z=x y+x fw→assign
Compute x zx+y zy and simplify the result to x y+z
Calculus palette: Partial-derivative operator
Context Panel: Evaluate and Display Inline
Context Panel: Expand≻Expand
x ∂∂ x Z+y ∂∂ y Z = x⁢y+f⁡yx−D⁡f⁡yx⁢yx+y⁢x+D⁡f⁡yx= expand 2⁢x⁢y+x⁢f⁡yx
But 2 x y+x f=2 x y+z−x y=x y+z. Because z must appear in the final result, during computation assignment is made to Z, not z, to avoid conflicts when making the final substitution.
Maple Solution - Coded
Define z=x y+x fw.
Z≔x y+x fw:
Apply the diff and simplify commands to evaluate the expression x zx+y zy
simplifyx diffZ,x+y diffZ,y = x⁢f⁡yx+2⁢y
But x f+2 y=x f+2 x y=z−x y+2 x y=z+x y. Because z must appear in the final result, during computation assignment is made to Z, not z, to avoid conflicts when making the final substitution.
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