How do you differentiate #x+y=xy#?
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To differentiate ( x + y = xy ), we will first rewrite the equation to solve for ( y ) in terms of ( x ). Then, we will use implicit differentiation to find the derivatives with respect to ( x ).
First, rewrite the equation as ( y = xy - x ).
Now, differentiate both sides of the equation with respect to ( x ), using the product rule on the right side:
[ \frac{{dy}}{{dx}} = x\frac{{dy}}{{dx}} + y - 1 ]
Now, solve for ( \frac{{dy}}{{dx}} ):
[ \frac{{dy}}{{dx}} - x\frac{{dy}}{{dx}} = y - 1 \ \frac{{dy}}{{dx}}(1 - x) = y - 1 \ \frac{{dy}}{{dx}} = \frac{{y - 1}}{{1 - x}} ]
Next, substitute the expression for ( y ) from the original equation:
[ \frac{{dy}}{{dx}} = \frac{{xy - x - 1}}{{1 - x}} ]
This is the derivative of ( y ) with respect to ( x ) for the given equation ( x + y = xy ).
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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.
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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