How do you differentiate #1/cos(x) = x/y^2y#?
Use implicit differentiation and algebra to get
This simplifies to
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To differentiate the equation ( \frac{1}{\cos(x)} = \frac{x}{y^2  y} ) with respect to (x), you would follow these steps:
 Differentiate both sides of the equation with respect to (x).
 Use the chain rule and the quotient rule where necessary.
 Solve for ( \frac{dy}{dx} ) or any other derivatives if required.
Let's go through the steps:

Differentiate the left side: [ \frac{d}{dx} \left( \frac{1}{\cos(x)} \right) = \frac{\sin(x)}{\cos^2(x)} ]

Differentiate the right side: [ \frac{d}{dx} \left( \frac{x}{y^2  y} \right) = \frac{(y^2  y) \cdot \frac{d}{dx}(x)  x \cdot \frac{d}{dx}(y^2  y)}{(y^2  y)^2} ]

Simplify and solve for ( \frac{dy}{dx} ) if needed.
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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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