# How do you differentiate #f(x)=(4x-5)/(3x+2)# using the quotient rule?

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To differentiate ( f(x) = \frac{4x - 5}{3x + 2} ) using the quotient rule, where ( u(x) = 4x - 5 ) and ( v(x) = 3x + 2 ), the derivative ( f'(x) ) is given by:

[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ]

where ( u'(x) ) and ( v'(x) ) are the derivatives of ( u(x) ) and ( v(x) ) respectively.

Using the quotient rule:

[ f'(x) = \frac{(4)(3x + 2) - (4x - 5)(3)}{(3x + 2)^2} ]

[ f'(x) = \frac{12x + 8 - 12x + 15}{(3x + 2)^2} ]

[ f'(x) = \frac{23}{(3x + 2)^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.

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