Question

how do I do this ~ :(?
Find y' if e^xy = x+y

Answer 1

you have
\[e^{x+y}=x+y\]
\[e^x\times e^y=x+y\]
Let's differentiate this, do you know product rule of differentiation ?

Answer 2

nope i forgot them ~ can u remind me ?

Answer 3

\[(f\times g)'=f'\times g +f\times g'\]
Can you use this on the left side, right side is simple differentiation

Answer 4

(e^z)(e^y)+(e^z)(e^y)=1+1 ??

Answer 5

if you are differentiating with respect to x, e^y term , you'll get
\[\frac{d}{dx } e^y=e^y \times \frac{dy}{dx}\]

Answer 6

im sorry ~ im so confused right now ~

Answer 7

Let me explain you, ok do you know differentiation?

Answer 8

yes ~ but i don't know how to differentiate functions with 2 variables

Answer 9

do u use implicit differentiation or do i first move a variable to one side?

Answer 10

We'll have to use implicit here, we have y as an exponential and a linear term, we can't separate out the equation.

Answer 11

ok~ so when u use implicit differentiation ~ do start by putting ln on both sides?
ln(e^xy)=ln(x+y)??

Answer 12

Yeah, we can do that but it's not required always.

Answer 13

Let's just start from what we have
\[e^{xy}=x+y\]
let's differentiate this
\[\frac{d}{dx} (e^x e^y)=\frac{d}{dx} (x+y)\]
\[e^xe^y+e^xe^y \frac {dy}{dx}=1+\frac {dy}{dx}\]
Do you get this?

Answer 14

i don't get the second step~

Answer 15

i bad with all the dy/dx ~ i usually just put y' or y"

Answer 16

\[\frac{dy}{dx}= y'\]

\[\frac{d}{dx} (e^x e^y)=\frac{d}{dx} (x+y)\]
\[e^y\frac{d}{dx} e^x+e^x\frac{d}{dx} e^y=1+\frac {dy}{dx}\]
\[e^y e^x+e^x e^y \frac {dy}{dx}=1+\frac{dy}{dx}\]
\[e^y e^x+e^x e^y y'=1+y'\]

Answer 17

Does this make sense?

Answer 18

yes :) ~ so is that the final answer ~? or we have to make y' to one side?