Question

use implicit differentiation to differentiate:
1+ye^(y)=x+e^(x)

Answer 1

Someone PLEASE help! I'm soo stuck right now!!!

Answer 2

Oh thank you!!!

Answer 3

March through the equation taking the derivative with respect to x of each term.

Answer 4

I understand that, but what's the derivative of ye^(y)? I get taht you need to use the product rule, but I get stuck at dy/dx(e^y)!

Answer 5

The right hand side is easy — what did you get there?

Answer 6

I got 1+e^x. THat wasn't a problem! It's the other side that's got me confused!

Answer 7

I just want to make sure that you've gotten it so I don't get an abuse claim about showing you the answer without giving you a chance to work it out.

Answer 8

So, that gives us
\[\frac{d}{dx}[1+ye^y] = 1+e^x\]What is the derivative of 1?

Answer 9

0. but how do you get e^x?

Answer 10

Excellent. So that leaves
\[\frac{d}{dx}[ye^y] = 1+e^x\]That's a product there on the left side, so we can use the product rule. Do you remember what that is?

Answer 11

y*(dy/dx(e^y))+e^y(dy/dx(y)) Right?

Answer 12

Let's let \[u=e^y\] and \[v=y\] then \[\frac{d}{dx}[uv]=v\frac{du}{dx}+u\frac{dv}{dx}\]

Answer 13

So I was right, right? y*(dy/dx(e^y)) + e^(y)*(dy/dx(y))

Answer 14

I'm not 100% convinced by the way you are writing it...

Answer 15

What do you mean?
y multiplied by dy/dx(e^y) + e^(y) mulitplied by dy/dx(y)

Answer 16

do you mean
\[y(\frac{dy}{dx}(e^y)) + e^y(\frac{dy}{dx}(y))\]

Answer 17

Exactly!! So how would I take the derivative of e^y?

Answer 18

use the chain rule...

Answer 19

the reason I don't like the way you wrote it is that it is unclear that you don't mean x(y) in some of those places, for example, and often in doing implicit differentiation you'll want to do things like y(x) to remind yourself that y is a function of x...

Answer 20

How would I use the chain rule on e^y?

Answer 21

Let \(u=y\) and that gives you
\[\frac{d}{dx}(e^y) = \frac{de^u}{du}\frac{du}{dx}\]

Answer 22

Sooo... e^y multiplied by dy/dx

Answer 23

you're going to end up with \[\frac{dy}{dx}\]as a factor in both terms on the left hand side.

Answer 24

Was I right?

Answer 25

What is your final answer?

Answer 26

y*(e^y*(dy/dx)) + (e^y)*(dy/dx)

Answer 27

Just for the left hand side!! The other side we already know!

Answer 28

Hey are you there? Help, please?

Answer 29

wut?

Answer 30

yeah, that looks right.

Answer 31

Okayy, so then I would isolate the dy/dx! But how?

Answer 32

\[ye^y\frac{dy}{dx}+e^y\frac{dy}{dx} = 1 + e^x\]You can factor the left side, can't you?

Answer 33

Take out e^y and dy/dx, right?

Answer 34

You could do that. At least the dy/dx, in any case.

Answer 35

So the final anwer should look like:
dy/dx=1+e^x/(ye^y +e^y)

Answer 36

That should give you \[\frac{dy}{dx} (ye^y+e^y) = 1+e^x\] or \[\frac{dy}{dx} = \frac{1+e^x}{e^y(y+1)}\]

or if you factored it more completely:
\[e^y\frac{dy}{dx}(1+y) = 1+e^x\]\[e^y\frac{dy}{dx}=\frac{1+e^x}{1+y} \]\[\frac{dy}{dx}=\frac{e^{-y}}{1+y}+\frac{e^xe^{-y}}{1+y} = \frac{e^{-y}}{1+y} + \frac{e^{x-y}}{1+y}\]

Answer 37

Be careful about writing divisions: you wrote dy/dx = 1+e^x/(ye^y+e^y) which equals
\[\frac{dy}{dx}=1+\frac{e^x}{ye^y+e^y}\] by the rules of operator precedence...

Answer 38

Perfect!! I understand Now! Thanks for all the help! You really cleared that up for me!! And the answer key they gave us has it written the way I gave it... Thanks again!! *gives cookie* :)

Answer 39

Well, the answer key is wrong if it wrote it that way, as the two are not equivalent. You need parentheses around the numerator:

dy/dx = (1+e^x)/(ye^y+e^y)

At \(x =2, y = 3\),

\[\frac{1+e^x}{ye^y+e^y} = \frac{1+e^2}{4e^3} \approx 0.104417\]

but written carelessly,
\[1+\frac{e^x}{ye^y+e^y} = 1+\frac{1}{4e} \approx 1.90197\]