How do you use the limit definition to find the derivative of #f(x)=1/(4x-3)#?

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Answer 1 · 2017-01-09T14:41:36

#f'(x) = lim_(Deltax->0) (Deltaf)/(Deltax) =(-4)/((4x-3)^2)#

The definition of the derivative of #f(x)# is:

#f'(x) = lim_(Deltax->0) (f(x+Deltax)-f(x))/(Deltax)= lim_(Deltax->0) (Deltaf)/(Deltax)#

Let us calculate the increment of #f(x)#:

#Deltaf = 1/(4(x+Deltax)-3) -1/(4x-3)#

#Deltaf = (4x-3-4x-4Deltax+3)/((4x+4Deltax-3)(4x-3))=(-4Deltax)/((4x+4Deltax-3)(4x-3))#

So:

#(Deltaf)/(Deltax) =(-4)/((4x+4Deltax-3)(4x-3))#

and

#lim_(Deltax->0) (Deltaf)/(Deltax) =(-4)/((4x-3)^2)#