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How do you differentiate #f(x) = x^3/sinx# using the quotient rule?

Answer 1

Scarlett Allison

#(x^2(3sinx-xcosx))/sin^2x#

The quotient rule states that:

#d/dx(u/v)=(vu'-uv')/v^2#

#u=x^3, v=sinx# #u'=3x^2, v'=cosx#

#d/dx(x^3/sinx)=(sinx*3x^2-x^3*cosx)/sin^2x#

#=(x^2(3sinx-xcosx))/sin^2x#

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Answer 2

Emilia Aguilar

To differentiate ( f(x) = \frac{x^3}{\sin(x)} ) using the quotient rule, you apply the formula:

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

Where ( u = x^3 ) and ( v = \sin(x) ).

[ u' = 3x^2 ] [ v' = \cos(x) ]

Apply the quotient rule:

[ \frac{d}{dx} \left( \frac{x^3}{\sin(x)} \right) = \frac{(3x^2)(\sin(x)) - (x^3)(\cos(x))}{(\sin(x))^2} ]

[ = \frac{3x^2\sin(x) - x^3\cos(x)}{\sin^2(x)} ]

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Answer from HIX Tutor

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.