How do you differentiate #f(x)=(14x)/(1+4x)# using the quotient rule?
Given:
As per the Quotient Rule,
thus:
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To differentiate ( f(x) = \frac{{1  4x}}{{1 + 4x}} ) using the quotient rule:

Identify ( u(x) ) as the numerator and ( v(x) ) as the denominator. ( u(x) = 1  4x ) and ( v(x) = 1 + 4x ).

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

Differentiate ( u(x) ) and ( v(x) ) with respect to ( x ): [ u'(x) = 4 ] [ v'(x) = 4 ]

Substitute into the quotient rule formula: [ f'(x) = \frac{{(1 + 4x)(4)  (1  4x)(4)}}{{(1 + 4x)^2}} ]

Simplify the expression: [ f'(x) = \frac{{4  16x  4 + 16x}}{{(1 + 4x)^2}} ] [ f'(x) = \frac{{8}}{{(1 + 4x)^2}} ]
Therefore, the derivative of ( f(x) = \frac{{1  4x}}{{1 + 4x}} ) with respect to ( x ) is ( f'(x) = \frac{{8}}{{(1 + 4x)^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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