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How do you implicitly differentiate #y= (x-y) e^(xy)-xy^2 #?

Answer 1

Isaiah Allen

#y'=(e^(xy)+y(x-y)e^(xy)-y^2)/(1+e^(xy)-x(x-y)e^(xy)+2xy)#

Differentiating with respect to #x#

#y'=(1-y')e^(xy)+(x-y)e^(xy)(y+xy')-y^2-2xyy'# so we get #y'(1+e^(xy)-x(x-y)e^(xy)+2xy)=e^(xy)+y(x-y)e^(xy)-y^2#

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Answer 2

Isabella Ackerman

To implicitly differentiate ( y = (x - y)e^{xy} - xy^2 ), you can follow these steps:

Differentiate each term of the equation with respect to ( x ).
Apply the product rule and chain rule where necessary.
Solve for ( \frac{dy}{dx} ) after differentiation.

Here are the steps:

Differentiating each term:
- For ( (x - y)e^{xy} ), use the product rule and chain rule.
- For ( -xy^2 ), differentiate term by term.
Apply the product rule and chain rule where necessary.
Solve for ( \frac{dy}{dx} ) after differentiation.

The final result will give you the implicit derivative ( \frac{dy}{dx} ).

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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.