# How do you implicitly differentiate #xy- 3yln(x-y)= 4/x#?

Differentiating both sides of

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To implicitly differentiate the given equation ( xy - 3y \ln(x-y) = \frac{4}{x} ), follow these steps:

- Differentiate each term of the equation with respect to ( x ).
- Apply the product rule for the term ( xy ).
- Apply the chain rule for the term ( -3y \ln(x-y) ).
- Simplify the resulting expression.

The implicit differentiation yields:

[ y + x \frac{dy}{dx} - 3 \ln(x-y) - 3y \left(\frac{1}{x-y} \frac{d(x-y)}{dx}\right) = -\frac{4}{x^2} ]

Simplify the expression and solve for ( \frac{dy}{dx} ):

[ x \frac{dy}{dx} - 3 \ln(x-y) + \frac{3y}{x-y} = -\frac{4}{x^2} - y \frac{1}{x-y} ]

[ \frac{dy}{dx} = \frac{-\frac{4}{x^2} - y \frac{1}{x-y} + 3 \ln(x-y)}{x} - \frac{3y}{x(x-y)} ]

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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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