Question
Use the Chain Rule to calculate the partial derivatives 9(w,y) = cos(r? + yP) , x = 20 3u,y = (4u + v) 89 Ou 89 Ov Express your answer in terms of the independent variables U, U
Use the Chain Rule to calculate the partial derivatives 9(w,y) = cos(r? + yP) , x = 20 3u,y = (4u + v) 89 Ou 89 Ov Express your answer in terms of the independent variables U, U
Answers
Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. $$ \frac{\partial f}{\partial u} ; f(x, y)=x^{2}+y^{2}, x=e^{u+v}, y=u+v $$
Okay, F is a function of U. And V. And U and V. R. Functions of X and Y. So if we want to find diva D partial that with respect to why we gotta do D F d u d u D Y. Do you have t v DVD y ourself F with respect to you times the partial of U. With respect to why? Plus the partial of F. With respect to me times the partial viv? With respect to why? Alright, so F is E to the U plus V. So the derivative of E to the U plus V is E. To the U plus V Times the derivative of you plus B with respect to you. Which is one times the derivative of U. With respect to Y zero plus D. F T V. That's E to the U plus B times the derivative of U plus V. Which is one times T V Dy, which is X. So X. E. To the U plus B.
Function of X and Y. And then X and Y are functions of V and W. So far as a function of X and Y. And there are functions of V and W. So if I want to find D R D V then I have to go D r D x D x T V Gr dy dy devi Okay. D r d x times dx tv plus D R D Y. Time's D Y D V. Okay. The derivative of our with respect to X. It's something squared. So it's derivative is to something To the one power times the derivative of the something which with respect to X is just one Times DX TV Novi in there. So plus time zero plus drd Why? So it's still something squared To something to the one times the derivative of the something which would be -2 time's D Y D V. So V is a variable and w is a constant. So like we cubed or be to the 10th from VW the To the W -1. So that's zero. So minus four X -2. Y W. V. to the W -1. Okay, D R D W. Okay, go down the other way here too. There here to there. The X. Oops, I should have said it out loud, I wouldn't have written it wrong. The R D X D X D W plus D R D Y. The Y D W. Okay, the rdx something swear to x minus two white. The one times the derivative of X and then D x D W is to W plus D R. D Y two x minus two Y. To the one touch minus two time's D Y D W. So now W. is the variable, so this is like three to the W. or 12 to the W. Okay, so the derivative of that is copied exactly. So be to the W Times the derivative of the exponent which is one times the Ellen of the base. Because it's not E. That's tricky. So for W times x minus two y minus four X -2 Y. B. To the w natural log of V.
Problem function of X. Y. And Z. And X. Y and Z are functions of R. S. And T. Possibly. Okay, so our diagram looks like this X. Y. Z. R. S. T. R. S. T. R. S. T. Okay, so there are actually three partials we could have taken near D. F. D. R. D. F DT and DD FDS but they only access for two so D. F. D. R. So we have to go, yeah. DF dx dx D R plus D. F. Dy Dy D R plus D. F dizzy dizzy dear. Okay, so DF dx is why The XDR is one plus D. F. D. Y. Is X. D. Y. D. R. is three T. DFD Z two, Z dizzy D. R. Zero. So why Plus three x. T. And then DF DT DF dx dx DT plus D. F. Dy dy DT plus DF dizzy dizzy DT so DF dx the same thing as it was still why the X. D. T uh minus two D. F. D. Y. Is X. Dy DT three are The FDZ two Z dizzy D. T. Zero again So -2 Y Plus three x. r.
With respect to S. And R. F. Is a function of X. Y. And Z. So let's make us a little drawing here and each of X. Y. And s. Are functions of S. And are all the really? X and Z. Aren't, but let's just say they are R. S. R. S. R. S. Okay. So if you're finding D. F. D. S then you have to go down every path that has an S. On it. So it's D. X. D. S. For at the first one start at the top. D. E. F. D. X. Time's D. X. D. S plus the F. Dy dy Ds plus TF dizzy, dizzy tedious. Okay sort alright here we go. DF dx is why D. X. D. S. Is to S plus DfD Y. Is X. The yds is to our plus D. F. D. Z. Is to Z dizzy D. S. O. There's no is there so zero? So to Y. S plus two X. R. Then if you wanted to have just X. And Y or just rns in there you would plug in as squared for X. And to R. S. For Y. Okay then D. F. D. R. Can now you have to go down the lines that have bars on them. So D. F. D. X. D. XDR. The F. D. Y. D. Y. D. R. DF dizzy dizzy dear the F. D. X D. X. D. R plus D. F. D. Y. D. Y. D. R plus D. F. Dizzy dizzy D. R. Okay so the DF dx is going to be the same as it was before? Okay. So it's why time's D XDR No. Are there so Time zero plus DFD Y. Which is X time's D Y D R. So to S there plus D f D Z two Z times. DZ D R two R. Okay, so that is two Xs plus two hopes for VR Yeah.