# How do you implicitly differentiate #-1=(x+y)^2-xy-e^(3x+7y) #?

I find this soln. of mine very easier than the one I provided earlier!

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Diff. both sides,

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To implicitly differentiate the given equation ( -1 = (x + y)^2 - xy - e^{3x + 7y} ), follow these steps:

- Differentiate both sides of the equation with respect to ( x ) using the chain rule and product rule where necessary.
- Express the derivative of ( y ) with respect to ( x ) as ( \frac{dy}{dx} ).
- Solve for ( \frac{dy}{dx} ) to find the implicit derivative.

Differentiating the equation with respect to ( x ):

[ \frac{d}{dx}(-1) = \frac{d}{dx}((x + y)^2 - xy - e^{3x + 7y}) ]

[ 0 = 2(x + y) \frac{d}{dx}(x + y) - y - x \frac{dy}{dx} - e^{3x + 7y}(3 + 7\frac{dy}{dx}) ]

[ 0 = 2(x + y)(1 + \frac{dy}{dx}) - y - x \frac{dy}{dx} - e^{3x + 7y}(3 + 7\frac{dy}{dx}) ]

[ 0 = 2(x + y) + 2(x + y)\frac{dy}{dx} - y - x \frac{dy}{dx} - 3e^{3x + 7y} - 7e^{3x + 7y}\frac{dy}{dx} ]

[ 0 = 2(x + y) + (2(x + y) - y - 3e^{3x + 7y})\frac{dy}{dx} - x \frac{dy}{dx} - 7e^{3x + 7y}\frac{dy}{dx} ]

[ 0 = 2(x + y) + (2x + 2y - y - 3e^{3x + 7y})\frac{dy}{dx} - x \frac{dy}{dx} - 7e^{3x + 7y}\frac{dy}{dx} ]

[ 0 = 2(x + y) + (2x + y - 3e^{3x + 7y})\frac{dy}{dx} - x \frac{dy}{dx} - 7e^{3x + 7y}\frac{dy}{dx} ]

[ 0 = 2x + 2y - 3e^{3x + 7y} - x \frac{dy}{dx} - 7e^{3x + 7y}\frac{dy}{dx} ]

[ (2y - 3e^{3x + 7y}) = (x + 7e^{3x + 7y})\frac{dy}{dx} ]

[ \frac{dy}{dx} = \frac{2y - 3e^{3x + 7y}}{x + 7e^{3x + 7y}} ]

Therefore, the implicit derivative of ( y ) with respect to ( x ) is ( \frac{2y - 3e^{3x + 7y}}{x + 7e^{3x + 7y}} ).

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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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