We have: #f(x) = (frac(3 x - 5)(4 x - 7))^(- 6)#
This function can be differentiated using a combination of the "quotient rule" and the "chain rule".
Let #u = frac(3 x - 5)(4 x - 7)# and #v = u^(- 6)#:
#Rightarrow f'(x) = u' cdot v'#
#Rightarrow f'(x) = (frac((4 x - 7) cdot frac(d)(dx)(3 x - 5) - (3 x - 5) cdot frac(d)(dx)(4 x - 7))((4 x - 7)^(2))) cdot (- 6 u^(- 7))#
#Rightarrow f'(x) = (frac((4 x - 7)(3) - (3 x - 5)(4))((4 x - 7)^(2))) cdot (- 6 u^(- 7)#
#Rightarrow f'(x) = (frac(12 x - 21 - 12 x + 20)((4 x - 7)^(2))) cdot (- 6 u^(- 7))#
#Rightarrow f'(x) = (frac(- 1)((4 x - 7)^(2))) cdot (- 6 u^(- 7))#
#Rightarrow f'(x) = frac(6 u^(- 7))((4 x - 7)^(2))#
Let's replace #u# with #frac(3 x - 5)(4 x - 7)#:
#Rightarrow f'(x) = frac(6(frac(3 x - 5)(4 x - 7))^(- 7))((4 x -7)^(2))#
#Rightarrow f'(x) = frac(6(frac((3 x - 5)^(- 7))((4 x - 7)^(- 7))))((4 x -7)^(2))#
#Rightarrow f'(x) = frac(6(3 x - 5)^(- 7))((4 x - 7)^(- 5))#
#Rightarrow f'(x) = frac(frac(6)((3 x - 5)^(7)))(frac(1)((4 x - 7)^(5)))#
#Rightarrow f'(x) = frac(6(4 x - 7)^(5))((3 x - 5)^(7))#
