How do you differentiate #f(x) = (−7 x^2 − 5)^8 (2 x^2 − 9)^9# ?

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Answer 1 · 2016-05-07T04:07:28

#color(red)(f' (x)=4x(7x^2+5)^7*(2x^2-9)^8(119x^2-207))#

For the given function #f(x)=(-7x^2-5)^8 (2x^2-9)^9#,
we are going to use the formula #d/dx(uv)=ud/dx(v)+v*d/dx(u)#

Let #u=(-7x^2-5)^8=(-1)^8(7x^2+5)^8=(7x^2+5)^8# and #v=(2x^2-9)^9#

#f' (x)=(7x^2+5)^8 d/dx(2x^2-9)^9+(2x^2-9)^9*d/dx(7x^2+5)^8#

#f' (x)=(7x^2+5)^8*9(2x^2-9)^(9-1)d/dx(2x^2-9)+(2x^2-9)^9*8(7x^2+5)^(8-1)d/dx(7x^2+5)#

#f' (x)=(7x^2+5)^8*9(2x^2-9)^8*(4x-0)+(2x^2-9)^9*8(7x^2+5)^(7)(14x+0)#

#f' (x)=(7x^2+5)^8*9(2x^2-9)^8*(4x)+(2x^2-9)^9*8(7x^2+5)^(7)(14x)#

Factoring common factors

#f' (x)=#
#4x(7x^2+5)^7*(2x^2-9)^8[(7x^2+5)*9+(2x^2-9)*2(14)]#

#f' (x)=4x(7x^2+5)^7*(2x^2-9)^8[(63x^2+45)+(56x^2-252)]#

#color(red)(f' (x)=4x(7x^2+5)^7*(2x^2-9)^8(119x^2-207))#

God bless....I hope the explanation is useful.