# How do you find the derivative of #(x^2+x)/sinx#?

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Use the quotient rule:

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To find the derivative of ( \frac{x^2+x}{\sin(x)} ), you can use the quotient rule, which states that for functions ( u(x) ) and ( v(x) ), the derivative of ( \frac{u(x)}{v(x)} ) is given by:

[ \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ]

Where ( u'(x) ) and ( v'(x) ) denote the derivatives of ( u(x) ) and ( v(x) ) respectively.

Let ( u(x) = x^2 + x ) and ( v(x) = \sin(x) ). Then:

[ u'(x) = 2x + 1 ] [ v'(x) = \cos(x) ]

Substituting these into the quotient rule formula:

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

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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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