Question

I'm stuck on a problem dealing with "e", and using Implicit Differentiation to find dy/dx. I've tried solving this one, but am not sure what the correct process is, if I'm not doing it right.. ?
The problem: e^(x/y) = x-y; Find dy/dx with Implicit Differentiation.
Any and all help is greatly appreciated!

Answer 1

@AryaVegapunk :)

Answer 2

You remember your (e^x)' derivative? :)

Answer 3

\[\large\rm e^{x/y}=x-y\]

Answer 4

it's just e^x, right?

Answer 5

Good. So same thing with e^(x/y), but apply chain rule:\[\large\rm \frac{d}{dx}e^{stuff}=e^{stuff}(stuff)'\]

Answer 6

\[\large\rm \frac{d}{dx}e^{x/y}=e^{x/y}\left(\frac{x}{y}\right)'\]You could apply quotient rule maybe, unless you prefer product rule.

Answer 7

So...
\[(e^{\frac{x}{y}})*[(1)(y^(-1) + x(-y^(-2)dy/dx] = (1-1(dy/dx))\]

Answer 8

Lols, I prefer Product Rule. And my laptop just died so I'm in my iPod touch, just FYI if I'm slower now.

Answer 9

\[\large\rm e^{x/y}\left(x\cdot y^{-1}\right)'=e^{x/y}\left[1\cdot y^{-1}-1x\cdot y^{-2}y'\right]\]So you went with product rule?
Ok cool.
Hmm I'm not sure about your last simplification though..

Answer 10

Just a note, when you do exponents in the equation tool,
you have to do ^{ } not ^( )

Answer 11

Is that right so far?

Answer 12

\[\large\rm e^{x/y}=x-y\]\[\large\rm e^{x/y}\left[y^{-1}-xy^{-2}y'\right]=1-y'\]Oh oh you were differentiating the right side,
ok great, looks good so far

Answer 13

The derivative of the right side, (x-y)? That's not (1-(dy/dx))?

Answer 14

oh boy you are slow XD haha np

Answer 15

Gotcha! Thanks!

Answer 16

Now you need to get all of your y' to the same side.

Answer 17

Yeah, the app refreshes like molasses..

Answer 18

Distribute the exponential,\[\large\rm e^{x/y}y^{-1}-e^{x/y}xy^{-2}y'=1-y'\]The one with the y', add it to the other side,
and also subtract 1 to the left side,
\[\large\rm e^{x/y}y^{-1}-1=e^{x/y}xy^{-2}y'-y'\]Factor our your y'\[\large\rm e^{x/y}y^{-1}-1=(e^{x/y}xy^{-2}-1)y'\]Then divide, ya? :d

Answer 19

Quotient rule woulda been a little nicer I think.
These negative exponents are so ugly >.<

Answer 20

e^(x/y)*y^(-1) - e^(x/y)*xy^(-2)(dy/dx) = 1 - dy/dx

Answer 21

Can I do anything more than this? :/