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.
<< Previous Example Section 4.3
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)