How do you differentiate #f(x)=5/(3x^2 + 1)^(1/3)  5x^2# using the quotient rule?
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To differentiate ( f(x) = \frac{5}{(3x^2 + 1)^{\frac{1}{3}}}  5x^2 ) using the quotient rule, follow these steps:

Identify ( u ) and ( v ) as follows: ( u = 5 ) ( v = (3x^2 + 1)^{\frac{1}{3}} )

Compute ( u' ) and ( v' ): ( u' = 0 ) ( v' = \frac{1}{3}(3x^2 + 1)^{\frac{2}{3}} \cdot 6x )

Apply the quotient rule: ( f'(x) = \frac{u'v  uv'}{v^2} )

Substitute the values into the formula: ( f'(x) = \frac{0 \cdot (3x^2 + 1)^{\frac{1}{3}}  5 \cdot \frac{1}{3}(3x^2 + 1)^{\frac{2}{3}} \cdot 6x}{(3x^2 + 1)^{\frac{2}{3}}} )

Simplify the expression: ( f'(x) = \frac{30x}{3(3x^2 + 1)} )

Further simplify if necessary: ( f'(x) = \frac{10x}{3x^2 + 1} )
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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