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Question: Function of a function: expressing derivatives of the former in terms of the derivatives of the latter?

Question:Function of a function: expressing derivatives of the former in terms of the derivatives of the latter?

MaPal93 145 Maple
differentiation implicit
August 09 2024
1

Function X__2(y1,y2) is a function of X__1(y1,y2). How do I express (implicitly) d(X__2)/d(y1) and d(X__2)/d(y2) in terms of d(X__1)/d(y1) and d(X__1)/d(y2) and perhaps of X__1 and X__2 themselves?

I illustrate what I mean with an example for X__1(y1,y2), where d(X__1)/d(y1) and d(X__1)/d(y2) are written in a relatively compact form in terms of X__1 itself:
 

> 

restart;
> 

X1 := RootOf((8*y__1^14 + 32*y__1^12*y__2^2 + 48*y__1^10*y__2^4 + 32*y__1^8*y__2^6 + 8*y__1^6*y__2^8 + 16*y__1^12 + 80*y__1^10*y__2^2 + 160*y__1^8*y__2^4 + 160*y__1^6*y__2^6 + 80*y__1^4*y__2^8 + 16*y__1^2*y__2^10)*_Z^10 + (40*y__1^13*y__2^2 + 120*y__1^11*y__2^4 + 120*y__1^9*y__2^6 + 40*y__1^7*y__2^8 + 16*y__1^13 + 176*y__1^11*y__2^2 + 544*y__1^9*y__2^4 + 736*y__1^7*y__2^6 + 464*y__1^5*y__2^8 + 112*y__1^3*y__2^10)*_Z^9 + (84*y__1^12*y__2^4 + 168*y__1^10*y__2^6 + 84*y__1^8*y__2^8 - 20*y__1^14 + 28*y__1^12*y__2^2 + 552*y__1^10*y__2^4 + 1288*y__1^8*y__2^6 + 1132*y__1^6*y__2^8 + 348*y__1^4*y__2^10 - 48*y__1^12 - 192*y__1^10*y__2^2 - 272*y__1^8*y__2^4 - 128*y__1^6*y__2^6 + 48*y__1^4*y__2^8 + 64*y__1^2*y__2^10 + 16*y__2^12)*_Z^8 + (88*y__1^11*y__2^6 + 88*y__1^9*y__2^8 - 80*y__1^13*y__2^2 + 56*y__1^11*y__2^4 + 960*y__1^9*y__2^6 + 1432*y__1^7*y__2^8 + 608*y__1^5*y__2^10 - 48*y__1^13 - 416*y__1^11*y__2^2 - 944*y__1^9*y__2^4 - 736*y__1^7*y__2^6 + 16*y__1^5*y__2^8 + 256*y__1^3*y__2^10 + 80*y__1*y__2^12)*_Z^7 + (40*y__1^10*y__2^8 - 128*y__1^12*y__2^4 + 168*y__1^10*y__2^6 + 928*y__1^8*y__2^8 + 632*y__1^6*y__2^10 + 12*y__1^14 - 192*y__1^12*y__2^2 - 1084*y__1^10*y__2^4 - 1472*y__1^8*y__2^6 - 340*y__1^6*y__2^8 + 432*y__1^4*y__2^10 + 180*y__1^2*y__2^12 + 48*y__1^12 + 144*y__1^10*y__2^2 + 112*y__1^8*y__2^4 - 48*y__1^6*y__2^6 - 96*y__1^4*y__2^8 - 32*y__1^2*y__2^10)*_Z^6 + (-92*y__1^11*y__2^6 + 228*y__1^9*y__2^8 + 368*y__1^7*y__2^10 + 32*y__1^13*y__2^2 - 384*y__1^11*y__2^4 - 1272*y__1^9*y__2^6 - 704*y__1^7*y__2^8 + 376*y__1^5*y__2^10 + 224*y__1^3*y__2^12 + 48*y__1^13 + 304*y__1^11*y__2^2 + 432*y__1^9*y__2^4 + 16*y__1^7*y__2^6 - 288*y__1^5*y__2^8 - 128*y__1^3*y__2^10)*_Z^5 + (-24*y__1^10*y__2^8 + 96*y__1^8*y__2^10 + 24*y__1^12*y__2^4 - 412*y__1^10*y__2^6 - 576*y__1^8*y__2^8 + 164*y__1^6*y__2^10 + 160*y__1^4*y__2^12 + 4*y__1^14 + 172*y__1^12*y__2^2 + 532*y__1^10*y__2^4 + 252*y__1^8*y__2^6 - 328*y__1^6*y__2^8 - 216*y__1^4*y__2^10 - 16*y__1^12 - 32*y__1^10*y__2^2 + 32*y__1^6*y__2^6 + 16*y__1^4*y__2^8)*_Z^4 + (-8*y__1^11*y__2^6 - 192*y__1^9*y__2^8 + 28*y__1^7*y__2^10 + 60*y__1^5*y__2^12 + 16*y__1^13*y__2^2 + 232*y__1^11*y__2^4 + 296*y__1^9*y__2^6 - 168*y__1^7*y__2^8 - 184*y__1^5*y__2^10 - 16*y__1^13 - 64*y__1^11*y__2^2 - 32*y__1^9*y__2^4 + 64*y__1^7*y__2^6 + 48*y__1^5*y__2^8)*_Z^3 + (-15*y__1^10*y__2^8 + 9*y__1^6*y__2^12 + 24*y__1^12*y__2^4 + 116*y__1^10*y__2^6 - 36*y__1^8*y__2^8 - 80*y__1^6*y__2^10 - 4*y__1^14 - 40*y__1^12*y__2^2 - 48*y__1^10*y__2^4 + 40*y__1^8*y__2^6 + 52*y__1^6*y__2^8)*_Z^2 + (12*y__1^11*y__2^6 - 4*y__1^9*y__2^8 - 16*y__1^7*y__2^10 - 8*y__1^13*y__2^2 - 24*y__1^11*y__2^4 + 8*y__1^9*y__2^6 + 24*y__1^7*y__2^8)*_Z - y__1^10*y__2^8 - y__1^8*y__2^10 - 4*y__1^12*y__2^4 + 4*y__1^8*y__2^8):

alias(X__1=X1);

X__2 := -y__1*y__2^2*X__1*(8*X__1^7*y__1^10 + 24*X__1^7*y__1^8*y__2^2 + 24*X__1^7*y__1^6*y__2^4 + 8*X__1^7*y__1^4*y__2^6 + 24*X__1^6*y__1^9*y__2^2 + 48*X__1^6*y__1^7*y__2^4 + 24*X__1^6*y__1^5*y__2^6 + 26*X__1^5*y__1^8*y__2^4 + 26*X__1^5*y__1^6*y__2^6 + 8*X__1^4*y__1^7*y__2^6 + 32*X__1^7*y__1^8 + 128*X__1^7*y__1^6*y__2^2 + 192*X__1^7*y__1^4*y__2^4 + 128*X__1^7*y__1^2*y__2^6 + 32*X__1^7*y__2^8 + 16*X__1^6*y__1^9 + 192*X__1^6*y__1^7*y__2^2 + 480*X__1^6*y__1^5*y__2^4 + 448*X__1^6*y__1^3*y__2^6 + 144*X__1^6*y__1*y__2^8 - 16*X__1^5*y__1^10 + 32*X__1^5*y__1^8*y__2^2 + 392*X__1^5*y__1^6*y__2^4 + 624*X__1^5*y__1^4*y__2^6 + 280*X__1^5*y__1^2*y__2^8 - 36*X__1^4*y__1^9*y__2^2 + 72*X__1^4*y__1^7*y__2^4 + 396*X__1^4*y__1^5*y__2^6 + 288*X__1^4*y__1^3*y__2^8 - 28*X__1^3*y__1^8*y__2^4 + 84*X__1^3*y__1^6*y__2^6 + 156*X__1^3*y__1^4*y__2^8 - 7*X__1^2*y__1^7*y__2^6 + 33*X__1^2*y__1^5*y__2^8 - 64*X__1^5*y__1^8 - 192*X__1^5*y__1^6*y__2^2 - 192*X__1^5*y__1^4*y__2^4 - 64*X__1^5*y__1^2*y__2^6 - 32*X__1^4*y__1^9 - 288*X__1^4*y__1^7*y__2^2 - 480*X__1^4*y__1^5*y__2^4 - 224*X__1^4*y__1^3*y__2^6 + 8*X__1^3*y__1^10 - 88*X__1^3*y__1^8*y__2^2 - 408*X__1^3*y__1^6*y__2^4 - 312*X__1^3*y__1^4*y__2^6 + 12*X__1^2*y__1^9*y__2^2 - 108*X__1^2*y__1^7*y__2^4 - 196*X__1^2*y__1^5*y__2^6 + 4*X__1^2*y__1^3*y__2^8 + 2*X__1*y__1^8*y__2^4 - 46*X__1*y__1^6*y__2^6 + 4*X__1*y__1^4*y__2^8 - y__1^7*y__2^6 + y__1^5*y__2^8 + 32*X__1^3*y__1^8 + 64*X__1^3*y__1^6*y__2^2 + 32*X__1^3*y__1^4*y__2^4 + 16*X__1^2*y__1^9 + 96*X__1^2*y__1^7*y__2^2 + 80*X__1^2*y__1^5*y__2^4 + 32*X__1*y__1^8*y__2^2 + 64*X__1*y__1^6*y__2^4 + 16*y__1^7*y__2^4)/(-8*X__1^7*y__1^11*y__2^2 - 24*X__1^7*y__1^9*y__2^4 - 24*X__1^7*y__1^7*y__2^6 - 8*X__1^7*y__1^5*y__2^8 - 32*X__1^6*y__1^10*y__2^4 - 64*X__1^6*y__1^8*y__2^6 - 32*X__1^6*y__1^6*y__2^8 - 50*X__1^5*y__1^9*y__2^6 - 50*X__1^5*y__1^7*y__2^8 - 28*X__1^4*y__1^8*y__2^8 + 32*X__1^8*y__1^10 + 160*X__1^8*y__1^8*y__2^2 + 320*X__1^8*y__1^6*y__2^4 + 320*X__1^8*y__1^4*y__2^6 + 160*X__1^8*y__1^2*y__2^8 + 32*X__1^8*y__2^10 + 160*X__1^7*y__1^9*y__2^2 + 640*X__1^7*y__1^7*y__2^4 + 960*X__1^7*y__1^5*y__2^6 + 640*X__1^7*y__1^3*y__2^8 + 160*X__1^7*y__1*y__2^10 - 8*X__1^6*y__1^12 - 24*X__1^6*y__1^10*y__2^2 + 328*X__1^6*y__1^8*y__2^4 + 1048*X__1^6*y__1^6*y__2^6 + 1056*X__1^6*y__1^4*y__2^8 + 352*X__1^6*y__1^2*y__2^10 - 24*X__1^5*y__1^11*y__2^2 - 48*X__1^5*y__1^9*y__2^4 + 400*X__1^5*y__1^7*y__2^6 + 848*X__1^5*y__1^5*y__2^8 + 424*X__1^5*y__1^3*y__2^10 - 38*X__1^4*y__1^10*y__2^4 - 54*X__1^4*y__1^8*y__2^6 + 274*X__1^4*y__1^6*y__2^8 + 290*X__1^4*y__1^4*y__2^10 - 44*X__1^3*y__1^9*y__2^6 - 32*X__1^3*y__1^7*y__2^8 + 104*X__1^3*y__1^5*y__2^10 - 27*X__1^2*y__1^8*y__2^8 + 15*X__1^2*y__1^6*y__2^10 - 64*X__1^6*y__1^10 - 256*X__1^6*y__1^8*y__2^2 - 384*X__1^6*y__1^6*y__2^4 - 256*X__1^6*y__1^4*y__2^6 - 64*X__1^6*y__1^2*y__2^8 - 256*X__1^5*y__1^9*y__2^2 - 768*X__1^5*y__1^7*y__2^4 - 768*X__1^5*y__1^5*y__2^6 - 256*X__1^5*y__1^3*y__2^8 + 16*X__1^4*y__1^12 + 32*X__1^4*y__1^10*y__2^2 - 408*X__1^4*y__1^8*y__2^4 - 848*X__1^4*y__1^6*y__2^6 - 424*X__1^4*y__1^4*y__2^8 + 48*X__1^3*y__1^11*y__2^2 + 48*X__1^3*y__1^9*y__2^4 - 352*X__1^3*y__1^7*y__2^6 - 352*X__1^3*y__1^5*y__2^8 + 54*X__1^2*y__1^10*y__2^4 - 150*X__1^2*y__1^6*y__2^8 + 22*X__1*y__1^9*y__2^6 - 30*X__1*y__1^7*y__2^8 - 2*y__1^8*y__2^8 + 32*X__1^4*y__1^10 + 96*X__1^4*y__1^8*y__2^2 + 96*X__1^4*y__1^6*y__2^4 + 32*X__1^4*y__1^4*y__2^6 + 96*X__1^3*y__1^9*y__2^2 + 192*X__1^3*y__1^7*y__2^4 + 96*X__1^3*y__1^5*y__2^6 - 8*X__1^2*y__1^12 - 8*X__1^2*y__1^10*y__2^2 + 104*X__1^2*y__1^8*y__2^4 + 104*X__1^2*y__1^6*y__2^6 - 16*X__1*y__1^11*y__2^2 + 48*X__1*y__1^7*y__2^6 - 8*y__1^10*y__2^4 + 8*y__1^8*y__2^6):


Synthetic representation of derivatives


derivatives for X__1 okay: written in terms of y__1, y__2, and X__1 itself.

> 

derX1_y1 := diff(X1, y__1):
derX1_y2 := diff(X1, y__2):

Diff('X__1(y__1,y__2)', y__1) = collect~(normal(eval(derX1_y1, X1 = 'X__1(y__1,y__2)')), 'X__1(y__1,y__2)');
Diff('X__1(y__1,y__2)', y__2) = collect~(normal(eval(derX1_y2, X1 = 'X__1(y__1,y__2)')), 'X__1(y__1,y__2)');


derivatives for X__2 not okay (X__2 is itself a function of X__1): how do I write them in terms of y__1, y__2, X__1 AND the derivatives for X__1 just found? What's the most compact way to express these two derivatives below?

> 

derX2_y1 := diff(X__2, y__1):
derX2_y2 := diff(X__2, y__2):

Diff('X__2(y__1,y__2)', y__1) = collect~(normal(eval(eval(derX2_y1, derX1_y1='Diff('X__1(y__1,y__2)', y__1)'), X__1 = 'X__1(y__1,y__2)')), 'X__1(y__1,y__2)');
Diff('X__2(y__1,y__2)', y__2) = collect~(normal(eval(derX2_y2, X__1 = 'X__1(y__1,y__2)')), 'X__1(y__1,y__2)');
> 

``

 

Download compact_derivatives.mw


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