Step-by-step explanation:
Let [tex]y(x)=f(x)g(x)h(x)[/tex] where
[tex]f(x) = x^4[/tex]
[tex]g(x)= (1 -2x^5)^6[/tex]
[tex]h(x)= (5 - 8x^3)^2[/tex]
so that
[tex]y(x) = x^4(1 -2x^5)^6(5 - 8x^3)^2[/tex]
Recall that the derivative of the product of functions is
[tex]y'(x)=f'(x)g(x)h(x)+f(x)g'(x)h(x)+f(x)g(x)h'(x)[/tex]
so taking the derivatives of the individual functions, we get
[tex]f'(x) = 4x^3[/tex]
[tex]g'(x) = 6(1 - 2x^5)^5(-10x^4)[/tex]
[tex]h'(x) = 2(5 - 8x^3)(-24x^2)[/tex]
So the derivative of y(x) is given by
[tex]y'(x) = 4x^3(1 -2x^5)^6(5 - 8x^3)^2 + x^4 6(1 -2x^5)^5(-10x^4)(5 - 8x^3)^2 + x^4(1 -2x^5)^6 2(5 - 8x^3)(-24x^2)[/tex]
or
[tex]y'(x) = 4x^3(1 -2x^5)^6(5 - 8x^3)^2[/tex]
[tex]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:- 60x^8(1 -2x^5)^5(5 - 8x^3)^2[/tex]
[tex]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:- 48x^6(1 -2x^5)^6 2(5 - 8x^3)[/tex]
