# How do you differentiate #(-x^2 -5x-4 )/ (sinx)# using the quotient rule?

the answer will be [(-2x-5)*sinx-(-x^2-5x-4)*cosx]/sin^2x

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To differentiate (-x^2 -5x-4) / (sinx) using the quotient rule, we apply the formula:

d/dx [f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)] / [g(x)]^2

Where f(x) = -x^2 - 5x - 4 and g(x) = sin(x).

Then, we find the derivatives:

f'(x) = -2x - 5 (derivative of -x^2 - 5x - 4 with respect to x) g'(x) = cos(x) (derivative of sin(x) with respect to x)

Substitute these derivatives into the quotient rule formula:

d/dx [(-x^2 -5x-4) / (sinx)] = [(sin(x) * (-2x - 5)) - (-x^2 -5x-4) * cos(x)] / [sin(x)]^2

Simplify the expression:

= [(sin(x) * (-2x - 5)) + (x^2 + 5x + 4) * 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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