How do you implicitly differentiate #-1=xytan(x/y) #?
Use the Product Rule, Chain Rule, and Quotient Rule.
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To implicitly differentiate the equation ( -1 = xy \tan(\frac{x}{y}) ), follow these steps:
- Differentiate both sides of the equation with respect to ( x ).
- Use the product rule for the term ( xy ).
- Use the chain rule for the term ( \tan(\frac{x}{y}) ).
- Solve for ( \frac{dy}{dx} ).
Differentiating ( -1 = xy \tan(\frac{x}{y}) ) with respect to ( x ):
[ \frac{d}{dx}(-1) = \frac{d}{dx}(xy \tan(\frac{x}{y})) ]
Using the product rule for ( xy ):
[ 0 = y\frac{dx}{dx} + x\frac{dy}{dx} + xy \frac{d}{dx}(\tan(\frac{x}{y})) ]
Simplify and apply the chain rule for ( \tan(\frac{x}{y}) ):
[ 0 = y + x\frac{dy}{dx} + xy \left( \sec^2(\frac{x}{y}) \frac{d}{dx}(\frac{x}{y}) \right) ]
Differentiating ( \frac{x}{y} ) with respect to ( x ):
[ \frac{d}{dx}(\frac{x}{y}) = \frac{1}{y} - \frac{x}{y^2} \frac{dy}{dx} ]
Substitute back into the equation:
[ 0 = y + x\frac{dy}{dx} + xy \left( \sec^2(\frac{x}{y}) \left( \frac{1}{y} - \frac{x}{y^2} \frac{dy}{dx} \right) \right) ]
Simplify:
[ 0 = y + x\frac{dy}{dx} + \frac{x}{y} \sec^2(\frac{x}{y}) - x^2 \sec^2(\frac{x}{y}) \frac{dy}{dx} ]
Isolate ( \frac{dy}{dx} ):
[ x\frac{dy}{dx} - x^2 \sec^2(\frac{x}{y}) \frac{dy}{dx} = -y - \frac{x}{y} \sec^2(\frac{x}{y}) ]
[ \frac{dy}{dx}(x - x^2 \sec^2(\frac{x}{y})) = -y - \frac{x}{y} \sec^2(\frac{x}{y}) ]
[ \frac{dy}{dx} = \frac{-y - \frac{x}{y} \sec^2(\frac{x}{y})}{x - x^2 \sec^2(\frac{x}{y})} ]
This is the implicit derivative of ( y ) with respect to ( x ).
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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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