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How do you differentiate #f(x)= x/(4x-2)# using the quotient rule?

Answer 1

William Abdalla

#f'(x)=-1/(2(2x-1)^2#

#f(x)=x/(4x-2)=1/2(x/(2x-1))#

To simplify the computation, we could take 2 as the denominator.

#f'(x)=d/(dx)[1/2(x/(2x-1))]=1/2d/(dx)(x/(2x-1))=1/2d/(dx)i(x)#

Now #i(x)=(h(x))/g(x)#

According to the Quotient Rule:

given: #i(x)=(h(x))/g(x)#

#i'(x)=(h'(x)*g(x)-g'(x)*h(x))/(g(x))^2#

#:.f'(x)=1/2[(1*(2x-1)-(2-0)*x)/(2x-1)^2]=#

#=1/2[(color(green)cancel(2x)-1color(green)cancel(-2x))/(2x-1)^2]=#

#=1/2*(-1/(2x-1)^2)=-1/(2(2x-1)^2#

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Answer 2

Emily Ammerman

To differentiate ( f(x) = \frac{x}{4x - 2} ) using the quotient rule:

Identify ( u(x) ) and ( v(x) ): ( u(x) = x ) and ( v(x) = 4x - 2 )
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) = 1 ) and ( v'(x) = 4 )
Substitute into the formula: ( f'(x) = \frac{(1)(4x - 2) - (x)(4)}{(4x - 2)^2} )
Simplify the expression: ( f'(x) = \frac{4x - 2 - 4x}{(4x - 2)^2} )
Further simplify: ( f'(x) = \frac{-2}{(4x - 2)^2} )

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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.