How do you differentiate #f(x)=(2x^24x1)/(x5)# using the quotient rule?
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To differentiate the function ( f(x) = \frac{{2x^2  4x  1}}{{x  5}} ) using the quotient rule, follow these steps:

Identify ( u(x) ) as the numerator and ( v(x) ) as the denominator. ( u(x) = 2x^2  4x  1 ) ( v(x) = x  5 )

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

Find the derivatives of ( u(x) ) and ( v(x) ): ( u'(x) = 4x  4 ) ( v'(x) = 1 )

Plug the derivatives and original functions into the quotient rule formula: ( f'(x) = \frac{{(4x  4)(x  5)  (2x^2  4x  1)(1)}}{{(x  5)^2}} )

Simplify the expression: ( f'(x) = \frac{{4x^2  20x  4x + 20  2x^2 + 4x + 1}}{{(x  5)^2}} ) ( f'(x) = \frac{{2x^2  24x + 21}}{{(x  5)^2}} )
Therefore, the derivative of ( f(x) ) is ( f'(x) = \frac{{2x^2  24x + 21}}{{(x  5)^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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