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.