# What is the implicit derivative of #13=2xe^y-x^3e^(2y) #?

Let's first find the derivative of each inner part.

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To find the implicit derivative of the given equation ( 13 = 2xe^y - x^3e^{2y} ), we differentiate both sides of the equation with respect to ( x ) using the chain rule and product rule.

Differentiating ( 13 ) with respect to ( x ) gives ( 0 ), and for the right-hand side, we have: [ \frac{d}{dx} (2xe^y) = 2e^y + 2xe^y \frac{dy}{dx} ] [ \frac{d}{dx} (x^3e^{2y}) = 3x^2e^{2y} + x^3(2e^{2y}) \frac{dy}{dx} ]

Putting it all together and solving for ( \frac{dy}{dx} ): [ 0 = 2e^y + 2xe^y \frac{dy}{dx} - (3x^2e^{2y} + 2x^3e^{2y} \frac{dy}{dx}) ] [ \frac{dy}{dx} (2xe^y - 2x^3e^{2y}) = 3x^2e^{2y} - 2e^y ] [ \frac{dy}{dx} = \frac{3x^2e^{2y} - 2e^y}{2xe^y - 2x^3e^{2y}} ]

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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.

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.

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.

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