How do you differentiate #f(x)=[(3x^2 + 1)^(1/3) - 5]^2 / [5x^2 + 4]^(1/2)# using the quotient rule?
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To differentiate ( f(x) = \frac{[(3x^2 + 1)^{\frac{1}{3}} - 5]^2}{\sqrt{5x^2 + 4}} ) using the quotient rule, follow these steps:
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Identify ( u ) and ( v ): ( u = (3x^2 + 1)^{\frac{1}{3}} - 5 )
( v = (5x^2 + 4)^{\frac{-1}{2}} ) -
Compute the derivatives of ( u ) and ( v ): ( u' = \frac{1}{3}(3x^2 + 1)^{-\frac{2}{3}}(6x) ) ( v' = -\frac{1}{2}(5x^2 + 4)^{-\frac{3}{2}}(10x) )
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Apply the quotient rule: ( f'(x) = \frac{u'v - uv'}{v^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.
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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