Chapter 4: Partial Differentiation
Section 4.3: Chain Rule
If z=y fx2−y2, show that y zx+x zy=x z/y.
It is most convenient to define wx,y=x2−y2 so that z=y fwx,y.
The following calculation then results from an application of the chain rule.
y zx+x zy
=y y f′wx+x y f′wy+f
=y2 f′2 x+x y f′−2 y+f
=2 x y2−2 x y2f′+x f
Maple Solution - Interactive
Define w and z
Context Panel: Assign Name
Compute xy zx+x zy and simplify the result to x z/y
Calculus palette: Partial-derivative operator
Context Panel: Evaluate and Display Inline
Context Panel: Simplify≻Simplify
y ∂∂ x Z+x ∂∂ y Z = 2⁢y2⁢D⁡f⁡x2−y2⁢x+x⁢f⁡x2−y2−2⁢y2⁢D⁡f⁡x2−y2= simplify f⁡x2−y2⁢x
But x f=xz/y=x z/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.
Maple Solution - Coded
Define z=y fw.
Apply the diff and simplify commands to evaluate the expression y zx+x zy
simplifyy diffZ,x+x diffZ,y = f⁡x2−y2⁢x
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