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How do you find the derivative of #e^y = xy ^2#?

Answer 1

Emily Ammerman

Differentiate implicitly:

#d/dx(e^y) = d/dx(xy^2)#

#e^y(dy/dx) = y^2 + 2xy(dy/dx)#

#e^y(dy/dx) - 2xy(dy/dx) = y^2#

#dy/dx(e^y - 2xy) = y^2#

#dy/dx = y^2/(e^y - 2xy)#

Hopefully this helps!

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Answer 2

Charles Anctil

To find the derivative of ( e^y = xy^2 ), you can use implicit differentiation:

Differentiate both sides of the equation with respect to ( x ).
Apply the chain rule to the left side.
Differentiate ( xy^2 ) with respect to ( x ) on the right side using the product rule.

The steps are as follows:

Differentiating ( e^y ) with respect to ( x ) using the chain rule gives ( e^y \cdot \frac{{dy}}{{dx}} ).
Applying the product rule to ( xy^2 ), we get ( x \cdot 2y \frac{{dy}}{{dx}} + y^2 ).
Combining these results, we have ( e^y \cdot \frac{{dy}}{{dx}} = x \cdot 2y \frac{{dy}}{{dx}} + y^2 ).
Solve for ( \frac{{dy}}{{dx}} ) by isolating it on one side of the equation. You'll get:

[ \frac{{dy}}{{dx}} = \frac{{y^2}}{{e^y - 2xy}} ]

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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.