How do you differentiate #f(x)= ( 4- secx )/ (x + 1) # using the quotient rule?
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To differentiate ( f(x) = \frac{4 - \sec(x)}{x + 1} ) using the quotient rule:
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Identify ( u(x) ) and ( v(x) ).
- Let ( u(x) = 4 - \sec(x) ) and ( v(x) = x + 1 ).
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Find ( u'(x) ) and ( v'(x) ).
- ( u'(x) = 0 - \sec(x) \tan(x) )
- ( v'(x) = 1 )
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Apply the quotient rule:
- ( f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} )
- Substitute the values: [ \begin{align*} f'(x) &= \frac{(x + 1)(0 - \sec(x) \tan(x)) - (4 - \sec(x))(1)}{(x + 1)^2} \ &= \frac{-x\sec(x)\tan(x) - \sec(x) + 1}{(x + 1)^2} \end{align*} ]
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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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