Here:
#d/dx x^5 * (x^2-3)^6#
we can use product rule:
#d/dx color(red)a * color(blue)b = (color(red)(a))'(color(blue)b) + (color(red)a)(color(blue)b)'#
So:
#d/dx color(red)(x^5) * color(blue)((x^2-3)^6)#
becomes:
#(color(red)x^5)'color(blue)((x^2-3)^6)+(color(red)x^5)(color(blue)((x^2-3)^6))'#
Simplifying:
#(5x^4)color(blue)((x^2-3)^6)+(color(red)x^5)(color(blue)((x^2-3)^6))'#
#d/dx color(blue)((x^2-3)^6)#
We can use chain rule here:
#d/dxf(x) = d/(du)f(u) * d/dx (x)#
#->d/dx(x^2-3)^6#
becomes:
#d/dx (u)^6 * d/dx (x^2-3)#
#=6u^5*2x#
Since #u=(x^2-3)#:
#=6(x^2-3)^5*2x#
#d/dx color(blue)((x^2-3)^6)=12x(x^2-3)^5#
Simplifying our former equation:
#(5x^4)color(blue)((x^2-3)^6)+(color(red)x^5)(color(blue)((x^2-3)^6))'#
becomes:
#(5x^4)(x^2-3)^6+(x^5)(12x)(x^2-3)^5#
Multiplying it out:
#=5x^4(x^2-3)^6+12x^6(x^2-3)^5#
And there we have our answer