# How do you differentiate #e^(x/y)=x-y#?

Prerequisites : The Usual Rules of Differentiation.

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To differentiate the equation ( e^{x/y} = x - y ), you can use implicit differentiation. The result will involve both partial derivatives. Taking the derivative of both sides with respect to ( x ) gives:

[ \frac{\partial}{\partial x} (e^{x/y}) = \frac{\partial}{\partial x} (x - y) ]

[ \frac{1}{y} e^{x/y} \frac{\partial}{\partial x} (x/y) = 1 ]

[ \frac{1}{y} e^{x/y} \left(\frac{\partial}{\partial x} \left(\frac{x}{y}\right)\right) = 1 ]

[ \frac{1}{y} e^{x/y} \left(\frac{1}{y}\right) = 1 ]

[ e^{x/y} \frac{1}{y^2} = 1 ]

[ e^{x/y} = y^2 ]

Now, if you want to find the partial derivative with respect to ( y ), you'll follow the same steps, but differentiating with respect to ( y ) instead of ( x ):

[ \frac{\partial}{\partial y} (e^{x/y}) = \frac{\partial}{\partial y} (x - y) ]

[ \frac{\partial}{\partial y} \left(\frac{x}{y}\right) = -1 ]

[ -\frac{x}{y^2} = -1 ]

[ \frac{x}{y^2} = 1 ]

[ x = y^2 ]

So, the equations derived from differentiating ( e^{x/y} = x - y ) with respect to ( x ) and ( y ) are ( e^{x/y} = y^2 ) and ( x = y^2 ), respectively.

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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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