SOLUTION: Can you help me with this problem? Solve by means of implicit differentiation: (e^x) + (e^y) = (e^(x+y)). Thanks tutors.

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Question 765977: Can you help me with this problem?
Solve by means of implicit differentiation:
(e^x) + (e^y) = (e^(x+y)).
Thanks tutors.
Found 2 solutions by solver91311, rothauserc:
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

Just another application of the sum rule and the chain rule.

Then second application of the sum and chain rules:

And the rest is just a little algebra:

Either of the last two forms are acceptable for an answer.

John

Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it

Answer by rothauserc(4718) (Show Source):
You can put this solution on YOUR website!
Solve by means of implicit differentiation:
(e^x) + (e^y) = (e^(x+y))
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recall that the derivative of e^x is e^x and the derivative of e^y is e^y*y'
e^x + e^y*y' = e^(x+y)*derivative of (x+y)
e^x +e^y*y' = e^(x+y)*(1+y')
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now solve for y'
e^y*y' = e^(x+y) +y'*e^(x+y)
e^y*y' - y'*e^(x+y) = e^(x+y) - e^x
y' = (e^(x+y) - e^x) / (e^y - e^(x+y))
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now e^(x+y) = e^x + e^y, so let's substitute
y' = (e^x + e^y - e^x) / (e^y - e^x - e^y)
y' = e^y / -e^x = -(e^(y-x))