You can differentiate this function by using the sum and power rules. Notice that you can rewrite this function as
y = (x^2 + x)^2 = [x(x+1)]^2 = x^2 * (x+1)^2
y = x^2 * (x^2 + 2x + 1) = x^4 + 2x^2 + x^2
Now, the sum rule tells you that for functions that take the form
y = sum_(i=1)^(oo)f_i(x)
you can find the derivative of y by adding the derivatives of those individual functions.
color(blue)(d/dx(y) = f_1^'(x) + f_2^'(x) + ...
In your case, you have
y^' = d/dx(x^4 + 2x^2 + x^2)
y^' = d/dx(x^4) + d/dx(2x^2) + d/dx(x^2)
y^' = d/dx(x^4) * 2d/dx(x^3) * d/dx(x^2)
To differentiate this fractions, use the power rule
color(blue)(d/dx(x^a) = ax^(a-1))
So, your derivative will come out to be
y^' = 4x^(4-1) + 2 * 3x^(3-1) + 2x^(2-1)
y^' = color(green)(4x^3 + 6x^2 + 2x)
Alternatively, you can use the chain rule to differentiate y.
color(blue)(d/dx(y) = d/(du)(y) * d/dx(u))
In your case, you have y = u^2 and u=x^2 + x, so that you get
dy/(dx) = d/(du)u^2 * d/dx(x^2 + x)
dy/dx = 2u * (2x + 1)
dy/dx = 2(x^2 + x) * (2x + 1)
dy/dx = (2x^2 + 2x) * (2x + 1)
dy/dx = 4x^3 + 2x^2 + 4x^2 + 2x = color(green)(4x^3 + 6x^2 + 2x)
