How do you differentiate #f(x) = (x)/(x^4-x^3+6)# using the quotient rule?
The quotient rule states that
Applying this to the given function, we see that
Both of these derivatives can be found through the power rule:
Simplified, this gives
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To differentiate ( f(x) = \frac{x}{x^4 - x^3 + 6} ) using the quotient rule, you apply the formula: ( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} ). Here, ( u = x ) and ( v = x^4 - x^3 + 6 ). Taking the derivatives, ( u' = 1 ) and ( v' = 4x^3 - 3x^2 ). Plugging these into the quotient rule formula, you get:
[ f'(x) = \frac{(x^4 - x^3 + 6) \cdot 1 - x \cdot (4x^3 - 3x^2)}{(x^4 - x^3 + 6)^2} ]
Simplify the expression:
[ f'(x) = \frac{x^4 - x^3 + 6 - (4x^4 - 3x^3)}{(x^4 - x^3 + 6)^2} ] [ f'(x) = \frac{-3x^4 + 4x^3 + 6}{(x^4 - x^3 + 6)^2} ]
That's the derivative of ( f(x) ) using the quotient rule.
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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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