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How do you differentiate #1=(xy)/(e^x+xy)#?

Answer 1

Isabella Ackerman

There is no derivative since the domain is empty.

Given:

#1 = (xy)/(e^x+xy)#

Multiply both sides by #e^x+xy# to get:

#e^x + xy = xy#

Subtract #xy# from both sides to get:

#e^x = 0#

There is no value of #x# which satisfies this and so the domain is empty.

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Answer 2

William Abdalla

To differentiate (1 = \frac{xy}{e^x + xy}) with respect to (x), you can use the quotient rule.

[\frac{d}{dx}\left(\frac{xy}{e^x + xy}\right) = \frac{(y(e^x + xy) - xy(e^x + 1))}{(e^x + xy)^2}]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.