Let u=6x2 and v=(x+5x2)3. Note that x+5x2=x(1+5x) so v=x3(1+5x)3.
du/dx=12x and dv/dx=3(x+5x2)2(1+10x)=3x2(1+5x)2(1+10x).
y=uv, so dy/dx=udv/dx+vdu/dx,
dy/dx=(6x2)(3x2(1+5x)2(1+10x))+(x3(1+5x)3)(12x),
dy/dx=18x4(1+5x)2(1+10x)+12x4(1+5x)3, which can be simplified by isolating the common factor 6x4(1+5x)2:
dy/dx=6x4(1+5x)2(3+30x+2+10x)=6x4(1+5x)2(5+40x)=30x4(1+5x)2(1+8x), because 5+40x=5(1+8x).
Another approach is to let u=6x5 and v=(1+5x)3.
du/dx=30x4 and dv/dx=15(1+5x)2,
dy/dx=6x5(15(1+5x)2)+(1+5x)3(30x4)=90x5(1+5x)2+30x4(1+5x)3,
dy/dx=30x4(1+5x)2(3x+1+5x)=30x4(1+5x)2(1+8x) as before.