How do you differentiate #f(x)= ( x +2sinx )/ (x + 4 )# using the quotient rule?
According to the quotient rule, a function's derivative can be expressed as the quotient of two other functions.
possesses a derivative of
This results in:
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To differentiate ( f(x) = \frac{x + 2\sin(x)}{x + 4} ) using the quotient rule, follow these steps:

Identify ( u(x) ) and ( v(x) ). ( u(x) = x + 2\sin(x) ) and ( v(x) = x + 4 ).

Compute ( u'(x) ) and ( v'(x) ). ( u'(x) = 1 + 2\cos(x) ) and ( v'(x) = 1 ).

Apply the quotient rule formula: [ f'(x) = \frac{v(x)u'(x)  u(x)v'(x)}{(v(x))^2} ]

Substitute the values: [ f'(x) = \frac{(x + 4)(1 + 2\cos(x))  (x + 2\sin(x))(1)}{(x + 4)^2} ]

Simplify the expression: [ f'(x) = \frac{x + 4 + 2x\cos(x) + 4\cos(x)  x  2\sin(x)}{(x + 4)^2} ]

Further simplify if needed: [ f'(x) = \frac{2x\cos(x) + 4\cos(x)  2\sin(x) + 4}{(x + 4)^2} ]
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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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