How do you differentiate #f(x)=x/(2x-2)-2/(2x-2)# using the quotient rule?

Answer 1

As they share the same denominator, we can add those fractions and then apply quotient rule to the function (now aggregated): #f(x)=(x-2)/(2x-2)#

Quotient rule states that for #y=f(x)/g(x)#, then #(dy)/(dx)=(f'(x)g(x)-f(x)g'(x))/g(x)^2#

So:

#(dy)/(dx)=(1(2x-2)-(x-2)(2))/(2x-2)^2=(2x-2-2x+4)/(2x-2)^2#

#(dy)/(dx)=2/(2x-2)^2=2/(4x^2-4x+4)=1/(2(x^2-x+1))#

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Answer 2

Charles Arocha

To differentiate the function ( f(x) = \frac{x}{2x - 2} - \frac{2}{2x - 2} ) using the quotient rule, follow these steps:

Identify the numerator and denominator functions: ( u(x) = x ) and ( v(x) = 2x - 2 ).
Apply the quotient rule: ( 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) = 1 ) and ( v'(x) = 2 ).
Substitute these values into the quotient rule formula.
Simplify the expression to obtain the derivative of ( f(x) ).

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Answer from HIX Tutor

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.