# How do you derive #y = (x-1)/sin(x)# using the quotient rule?

The derivative is

The rule states that

Now we have all the "ingredients", and we can put them in the "recipe":

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To derive ( y = \frac{x-1}{\sin(x)} ) using the quotient rule, follow these steps:

- Identify ( u = x - 1 ) and ( v = \sin(x) ).
- Apply the quotient rule formula: ( \frac{d}{dx} \left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} ).
- Differentiate ( u ) and ( v ) with respect to ( x ).
- Substitute the derivatives and the original functions into the quotient rule formula.
- Simplify the expression if possible.

Following these steps, you'll obtain the derivative of ( y ) with respect to ( x ).

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To derive (y = \frac{x - 1}{\sin(x)}) using the quotient rule, where (u = x - 1) and (v = \sin(x)), we apply the quotient rule formula:

[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} ]

Here, (\frac{du}{dx}) is the derivative of (u) with respect to (x) and (\frac{dv}{dx}) is the derivative of (v) with respect to (x).

First, let's find (\frac{du}{dx}) and (\frac{dv}{dx}):

(\frac{du}{dx} = \frac{d}{dx}(x - 1) = 1)

(\frac{dv}{dx} = \frac{d}{dx}(\sin(x)) = \cos(x))

Now, plug these into the quotient rule formula:

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

Simplify this expression to get the derivative.

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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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