Using the chain rule in conjunction with the product rule, we get:
dy/dx [ (x^3 + 2)^2(x^5 + 4)^4]dydx[(x3+2)2(x5+4)4]
=d/dx[(x^3+2)^2]⋅(x^5+4)^4+(x^3+2)2⋅d/dx[(x^5+4)^4]=ddx[(x3+2)2]⋅(x5+4)4+(x3+2)2⋅ddx[(x5+4)4]
=2(x^3+2)⋅d/dx[x^3+2]⋅(x^5+4)^4+4(x^5+4)^3⋅d/dx[x^5+4]⋅(x^3+2)^2=2(x3+2)⋅ddx[x3+2]⋅(x5+4)4+4(x5+4)3⋅ddx[x5+4]⋅(x3+2)2
=2(d/dx[x^3]+d/dx[2])(x^3+2)(x^5+4)^4+4(d/dx[x^5]+d/dx[4])(x3+2)^2(x^5+4)^3=2(ddx[x3]+ddx[2])(x3+2)(x5+4)4+4(ddx[x5]+ddx[4])(x3+2)2(x5+4)3
=2(3x^2+0)(x^3+2)(x^5+4)^4+4(5x^4+0)(x^3+2)^2(x^5+4)^3=2(3x2+0)(x3+2)(x5+4)4+4(5x4+0)(x3+2)2(x5+4)3
=6x^2(x^3+2)(x^5+4)^4+20x^4(x^3+2)^2(x^5+4)^3=6x2(x3+2)(x5+4)4+20x4(x3+2)2(x5+4)3
Which, when simplified, is equal to:
=2x^2(x^3+2)(x^5+4)^3(13^x5+20x^2+12)=2x2(x3+2)(x5+4)3(13x5+20x2+12)