Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

calyne

  • 2 years ago

Find dy/dx by implicit differentiation: e^(x/y) = x-y

  • This Question is Closed
  1. .Sam.
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    e^(x/y) = x-y e^(xy^(-1)) = x-y differentiate, \[[-\frac{x}{y^2}\frac{dy}{dx}+\frac{1}{y}]e^{xy^{-1}}=1-\frac{dy}{dx}\] \[\frac{dy}{dx}(1-\frac{xe^{xy^{-1}}}{y^2})=1-\frac{e^{xy^{-1}}}{y}\] \[\huge \frac{dy}{dx}=\frac{1-\frac{e^{xy^{-1}}}{y}}{(1-\frac{xe^{xy^{-1}}}{y^2})}\]

  2. .Sam.
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    multiply y^2 to top and bottom and simplify

  3. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[e^{\frac{x}{y}}\times \frac{y-xy'}{y^2}=1-y'\] is a start

  4. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    or use .sam. method either way

  5. .Sam.
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    \[\text{Final result}\] \[\frac{dy}{dx}=\frac{y \left(e^{x/y}-y\right)}{x e^{x/y}-y^2}\]

  6. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @calyne is it clear that the right hand side becomes \[1-y'\]?

  7. calyne
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yeah thanks guys

  8. calyne
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    wait sam how did you get from \[[−xy2dydx+1y]exy−1=1−dydx dydx(1−xexy−1y2)=1−exy−1y\]

  9. calyne
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    wait flutter i mean how did you get from the first equation to the second in your first post

  10. .Sam.
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    There's 2 steps there, 1)multiply e^{xy^{-1}} into [−xy2dydx+1y] 2)moving the dy/dx from RHS to LHS then factor dy/dx

  11. .Sam.
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    \[[-\frac{x}{y^2}\frac{dy}{dx}+\frac{1}{y}]e^{xy^{-1}}=1-\frac{dy}{dx}\] \[-\frac{xe^{xy^{-1}}}{y^2}\frac{dy}{dx}+\frac{e^{xy^{-1}}}{y}+\frac{dy}{dx}=1\] \[\frac{dy}{dx}(1-\frac{xe^{xy^{-1}}}{y^2})=1-\frac{e^{xy^{-1}}}{y}\] \[\huge \frac{dy}{dx}=\frac{1-\frac{e^{xy^{-1}}}{y}}{(1-\frac{xe^{xy^{-1}}}{y^2})}\]

  12. calyne
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    can you show me how to equate that to [ y * ( y - e^(x/y) ) ] / [ y^2 - xe^(x/y) ] ??

  13. .Sam.
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    i got to go , bump this up for others to solve

  14. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    if we can start with \[e^{\frac{x}{y}}\times \frac{y-xy'}{y^2}=1-y'\] we can do it step by step, it is essentially algebra from here on in

  15. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    i would a) multiply both sides by \(y^2\) b) put everything with a \(y'\) on one side and everything else on the other c) factor \(y'\) out of the terms d) divide

  16. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[e^{\frac{x}{y}}\times \frac{y-xy'}{y^2}=1-y'\] \[e^{\frac{x}{y}}(y-xy')=y^2-y^2y'\] \[ye^{\frac{x}{y}}-xe^{\frac{x}{y}}y'=y^2-y^2y'\] \[y^2y'-xe^{\frac{x}{y}}y'=y^2-ye^{\frac{x}{y}}\] \[y'(y^2-xe^{\frac{x}{y}})=y^2-ye^{\frac{x}{y}}\]

  17. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    check my algebra because it is hard to write all this here, but that is the idea

  18. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    last step is to divide and you are done!

  19. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.