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
