Chapter 4: Partial Differentiation
Section 4.3: Chain Rule
If Fx,y is the composition of Gu,v with u=x2−y2, v=2 x y, obtain Fx and Fy in terms of Gu and Gv.
=Gu ux+Gv vx
=Gu uy+Gv vy
=Gu 2 x+Gv 2 y
=Gu −2 y+Gv 2 x
=2x Gu+y Gv
=2x Gv−y Gu
Maple Solution - Interactive
Calculus palette: Partial-differential operator
Context Panel: Evaluate and Display Inline
∂∂ x Gx2−y2,2 x y = 2⁢D1⁡G⁡x2−y2,2⁢x⁢y⁢x+2⁢D2⁡G⁡x2−y2,2⁢x⁢y⁢y
∂∂ y Gx2−y2,2 x y = −2⁢D1⁡G⁡x2−y2,2⁢x⁢y⁢y+2⁢D2⁡G⁡x2−y2,2⁢x⁢y⁢x
Maple Solution - Coded
Simplified Maple notation is available if the commands to the right are first executed.
Maple's representation of Fx and Fy
Implement the chain rule
Restore the variables x and y.
Assign ux,y and vx,y to the names u and v, respectively.
u≔x2−y2:v≔2 x y:
diffGu,v,x = 2⁢D1⁡G⁡x2−y2,2⁢x⁢y⁢x+2⁢D2⁡G⁡x2−y2,2⁢x⁢y⁢y
diffGu,v,y = −2⁢D1⁡G⁡x2−y2,2⁢x⁢y⁢y+2⁢D2⁡G⁡x2−y2,2⁢x⁢y⁢x
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