# How do you differentiate #f(x)=x/sinx# using the quotient rule?

Apply the rule:

This can, of course, be written in other ways using algebra and trigonometry.

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To differentiate ( f(x) = \frac{x}{\sin(x)} ) using the quotient rule, follow these steps:

- Identify the numerator and denominator of the function. In this case, the numerator is ( x ) and the denominator is ( \sin(x) ).
- Apply the quotient rule formula: [ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2} ] where ( u ) is the numerator and ( v ) is the denominator.
- Differentiate the numerator (( u' )) and the denominator (( v' )) separately.
- Substitute the derivatives into the quotient rule formula.
- Simplify the expression if possible.

Using the quotient rule: [ f'(x) = \frac{(\sin(x) \cdot 1) - (x \cdot \cos(x))}{(\sin(x))^2} ]

Simplify the expression: [ f'(x) = \frac{\sin(x) - x \cdot \cos(x)}{\sin^2(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.

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