How do you find the derivative of #f(x) = (2x-3) ^ -2#?

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Answer 1 · 2016-06-26T11:50:34

#= - 1/(2(2x-3)) #

#int \ (2x-3) ^ -2 \ dx#

the usual rule is #int \ x^n \ dx = (x^(n+1))/(n+1)#

here we have

#int \ (2x-3) ^ -2 \ dx#

with sub #z = 2x-3# to make it more transparent, and so that #dz = 2 dx# the integral becomes

#1/2 int \ z ^ -2 \ dz#

#= 1/2 int \ (z ^( (-2 + 1)))/(-2 + 1) \ dz#

#= 1/2 \ (z ^( -1))/(-1) #

#= -1/2 1/z #

#= - 1/(2(2x-3)) #