How do you differentiate #f(x)= (3x^24 )/ (2x 1 )# using the quotient rule?
Derivative of
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To differentiate the function ( f(x) = \frac{3x^2  4}{2x  1} ) using the quotient rule, follow these steps:

Identify the numerator function ( u(x) = 3x^2  4 ) and the denominator function ( v(x) = 2x  1 ).

Apply the quotient rule formula: [ \frac{d}{dx} \left(\frac{u(x)}{v(x)}\right) = \frac{v(x) \cdot u'(x)  u(x) \cdot v'(x)}{(v(x))^2} ]

Find the derivatives of ( u(x) ) and ( v(x) ): ( u'(x) = 6x ) (derivative of (3x^2  4)) ( v'(x) = 2 ) (derivative of (2x  1))

Substitute the derivatives and original functions into the quotient rule formula: [ f'(x) = \frac{(2x  1) \cdot (6x)  (3x^2  4) \cdot 2}{(2x  1)^2} ]

Simplify the expression: [ f'(x) = \frac{(12x^2  6x)  (6x^2  8)}{(2x  1)^2} ] [ f'(x) = \frac{12x^2  6x  6x^2 + 8}{(2x  1)^2} ] [ f'(x) = \frac{6x^2  6x + 8}{(2x  1)^2} ]

Further simplify if necessary.
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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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