# How do you differentiate #1/cos(x) = x/(x-y^2-y)#?

First put everything on one side

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To differentiate the given equation ( \frac{1}{\cos(x)} = \frac{x}{x - y^2 - y} ) with respect to ( x ), we use the quotient rule:

[ \frac{d}{dx} \left( \frac{1}{\cos(x)} \right) = \frac{(x - y^2 - y) \frac{d}{dx}(1) - 1 \cdot \frac{d}{dx}(\cos(x))}{(x - y^2 - y)^2} ]

Differentiating ( \frac{1}{\cos(x)} ) yields ( \frac{\sin(x)}{\cos^2(x)} ) and ( \frac{d}{dx}(\cos(x)) = -\sin(x) ).

Plugging these derivatives back into the quotient rule formula:

[ \frac{(x - y^2 - y) \cdot 0 - 1 \cdot (-\sin(x))}{(x - y^2 - y)^2} = \frac{\sin(x)}{\cos^2(x)} \cdot \frac{1}{x - y^2 - y} ]

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