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

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To differentiate the function ( f(x) = \frac{4 - 3 \sec(x)}{x - 3} ) using the quotient rule, you would follow these steps:

- Identify ( u(x) ) as the numerator function and ( v(x) ) as the denominator function.
- Apply the quotient rule formula: ( [u(x) \cdot v'(x) - v(x) \cdot u'(x)] / [v(x)]^2 ).
- Find the derivative of ( u(x) ) and ( v(x) ) separately.
- Substitute the derivatives and original functions into the quotient rule formula.
- Simplify the expression if possible.

The derivatives needed are:

- ( u'(x) = \frac{d}{dx}(4 - 3\sec(x)) )
- ( v'(x) = \frac{d}{dx}(x - 3) )

After finding ( u'(x) ) and ( v'(x) ), you would then substitute these values into the quotient rule formula:

[ f'(x) = \frac{(4 - 3\sec(x)) \cdot v'(x) - (x - 3) \cdot ( -3\sec(x)\tan(x))}{(x - 3)^2} ]

This yields the derivative of the function ( f(x) ) with respect to ( x ).

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To differentiate the function ( f(x) = \frac{4 - 3 \sec(x)}{x - 3} ) using the quotient rule, you apply the formula:

[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} ]

Where: ( u = 4 - 3 \sec(x) ) ( v = x - 3 )

( u' = -3\sec(x)\tan(x) ) ( v' = 1 )

Substituting these values into the quotient rule formula, you get:

[ f'(x) = \frac{(0 - (4 - 3\sec(x))(1))((x - 3)) - (4 - 3\sec(x))(1)}{(x - 3)^2} ]

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

[ f'(x) = \frac{-x + 3 + 3x\sec(x) - 9\sec(x)}{(x - 3)^2} ]

[ f'(x) = \frac{-x + 3 + 3x\sec(x) - 9\sec(x)}{(x - 3)^2} ]

[ f'(x) = \frac{-x + 3 + \sec(x)(3x - 9)}{(x - 3)^2} ]

So, the derivative of ( f(x) = \frac{4 - 3 \sec(x)}{x - 3} ) using the quotient rule is:

[ f'(x) = \frac{-x + 3 + \sec(x)(3x - 9)}{(x - 3)^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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