How do you find the derivative of # x ln y  y ln x = 1#?
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To find the derivative of ( x \ln(y)  y \ln(x) = 1 ), follow these steps:

Implicitly differentiate both sides of the equation with respect to ( x ).

Apply the product rule to the terms involving ( x ) and ( y ).

After differentiation, solve for ( \frac{dy}{dx} ) (the derivative of ( y ) with respect to ( x )).
Here's the detailed process:

Differentiate both sides of the equation with respect to ( x ): [ \frac{d}{dx}(x \ln(y)  y \ln(x)) = \frac{d}{dx}(1) ]

Apply the product rule for differentiation: [ \frac{d}{dx}(x \ln(y))  \frac{d}{dx}(y \ln(x)) = 0 ]

Use the product rule: [ \frac{d}{dx}(x \ln(y)) = \ln(y) \frac{dx}{dx} + x \frac{d}{dx}(\ln(y)) ] [ = \ln(y) + x \cdot \frac{1}{y} \frac{dy}{dx} ]
[ \frac{d}{dx}(y \ln(x)) = \ln(x) \frac{dy}{dx} + y \frac{d}{dx}(\ln(x)) ] [ = \ln(x) \frac{dy}{dx} + y \cdot \frac{1}{x} ]

Substitute back into the original equation: [ \ln(y) + \frac{x}{y} \frac{dy}{dx}  \ln(x) \frac{dy}{dx}  \frac{y}{x} = 0 ]

Rearrange terms and combine like terms: [ \ln(y)  \ln(x) + \frac{x}{y} \frac{dy}{dx}  \frac{y}{x} = 0 ]

Combine the logarithms: [ \ln\left(\frac{y}{x}\right) + \frac{x}{y} \frac{dy}{dx}  \frac{y}{x} = 0 ]

Solve for ( \frac{dy}{dx} ): [ \frac{x}{y} \frac{dy}{dx} = \frac{y}{x}  \ln\left(\frac{y}{x}\right) ] [ \frac{dy}{dx} = \frac{y}{x} \cdot \frac{x}{y}  \frac{1}{y} \ln\left(\frac{y}{x}\right) ] [ \frac{dy}{dx} = 1  \frac{1}{y} \ln\left(\frac{y}{x}\right) ]
Therefore, the derivative of ( x \ln(y)  y \ln(x) = 1 ) with respect to ( x ) is ( \frac{dy}{dx} = 1  \frac{1}{y} \ln\left(\frac{y}{x}\right) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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