# How do you differentiate #(x)/ (1-cosx)# using the quotient rule?

The quotient rule states that

We can find each of the internal derivatives and then plug them back in:

Plugging it all back in:

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To differentiate the function ( \frac{x}{1-\cos(x)} ) using the quotient rule:

- Let ( u = x ) and ( v = 1 - \cos(x) ).
- Apply the quotient rule: [ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} ]
- Compute ( \frac{du}{dx} ) and ( \frac{dv}{dx} ). [ \frac{du}{dx} = 1 ] [ \frac{dv}{dx} = \sin(x) ]
- Substitute the values into the quotient rule formula: [ \frac{d}{dx} \left( \frac{x}{1-\cos(x)} \right) = \frac{(1-\cos(x))(1) - (x)(\sin(x))}{(1-\cos(x))^2} ]
- Simplify the expression if necessary.

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