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How do you implicitly differentiate #y= xy^2 + x ^2 e^(4x+y) #?

Answer 1

Emma Aldridge

See answer below:

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Answer 2

Christopher Agresta

To implicitly differentiate ( y = xy^2 + x^2 e^{4x+y} ):

Differentiate each term with respect to ( x ).
Apply the product rule for the term ( xy^2 ).
Apply the product rule and chain rule for the term ( x^2 e^{4x+y} ).
Solve for ( \frac{dy}{dx} ) after differentiation.

The steps are as follows:

Differentiate each term:

( \frac{dy}{dx} = y^2 + 2xy \frac{dy}{dx} + 2xe^{4x+y} + x^2 \left(4e^{4x+y} + e^{4x+y} \frac{dy}{dx}\right) )
Rearrange terms involving ( \frac{dy}{dx} ):

( \frac{dy}{dx} - 2xy \frac{dy}{dx} - x^2 e^{4x+y} \frac{dy}{dx} = y^2 + 2xe^{4x+y} + 4x^2 e^{4x+y} )
Factor out ( \frac{dy}{dx} ):

( \frac{dy}{dx}(1 - 2xy - x^2 e^{4x+y}) = y^2 + 2xe^{4x+y} + 4x^2 e^{4x+y} )
Solve for ( \frac{dy}{dx} ):

( \frac{dy}{dx} = \frac{y^2 + 2xe^{4x+y} + 4x^2 e^{4x+y}}{1 - 2xy - x^2 e^{4x+y}} )

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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.