Question

x(x^2+1)^3 using the chain rule

Answer 1

We want d/dx [x (x^2+1)^3]. Use the product rule and the chain rule.

f(x) = x, g(x) = (x^2+1)^3
f' = 1, g' = 3(x^2+1)^2 * 2x = 6x(x^2+1)^2

So
d/dx [x (x^2+1)^3] = 1*(x^2+1)^3 + x * 6x(x^2+1)^2
= (x^2+1)^3 + 6x^2(x^2+1)^2
= (x^2+1)^2[(x^2+1) + 6x^2]
= (x^2+1)^2(7x^2+1).