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

Answer 1

# dy/dx={x-(x-y)ln(x-y)}/y, or, x/y{1-1/y*e^(x/y)}#.

Prerequisites : The Usual Rules of Differentiation.

Given that, #e^(x/y)=x-y#.
#:. lne^(x/y)=ln(x-y)#.
#:. x/y=ln(x-y), or, x=yln(x-y)#.
#:. d/dx{x}=d/dx{xln(x-y)}#.
#:.1=xd/dx{ln(x-y)}+ln(x-y)*d/dx{y},#
# i.e., 1=x*1/(x-y)*d/dx{x-y}+ln(x-y)*dy/dx,#
# or, 1=x/(x-y){d/dx(x)-d/dx(y)}+ln(x-y)*dy/dx#.
#:. 1=x/(x-y){1-dy/dx}+ln(x-y)*dy/dx#.
#:. 1=x/(x-y)-x/(x-y)*dy/dx+ln(x-y)dy/dx#.
#:. 1-x/(x-y)={ln(x-y)-x/(x-y)}dy/dx#.
#:. {(x-y)-x}/cancel(x-y)={(x-y)ln(x-y)-x}/cancel(x-y)*dy/dx#.
#:. -y={(x-y)ln(x-y)-x}dy/dx#.
# rArr dy/dx={x-(x-y)ln(x-y)}/y#.
Since, #(x-y)=e^(x/y) and ln(x-y)=x/y#, we may write,
#dy/dx=x/y-1/y*e^(x/y)*x/y=x/y{1-1/y*e^(x/y)}#.
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Answer 2

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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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.

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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