what does this derivative mean?

(d/dx)(d/dt)(x^2t + t^2x - 2x^3)

Or how do I take the derivative of that?

The expression (d/dx)(d/dt)(x^2t + t^2x - 2x^3) represents a mixed partial derivative. It is the derivative of the given expression with respect to x, followed by the derivative with respect to t.

To compute this derivative, we can apply the chain rule twice. First, let's differentiate with respect to x:

(d/dx)(x^2t + t^2x - 2x^3)

To differentiate x^2t with respect to x, we treat t as a constant and use the power rule:
(d/dx)(x^2t) = 2xt
The derivative of t^2x with respect to x treats t as a constant, and again applies the power rule:
(d/dx)(t^2x) = t^2

The derivative of -2x^3 with respect to x is found using the power rule:
(d/dx)(-2x^3) = -6x^2

Putting it all together, the derivative with respect to x is:
2xt + t^2 - 6x^2

Now, let's differentiate this result with respect to t:

(d/dt)(2xt + t^2 - 6x^2)

To differentiate 2xt with respect to t, we again treat x as a constant and apply the product rule:
(d/dt)(2xt) = 2x

The derivative of t^2 with respect to t is found using the power rule:
(d/dt)(t^2) = 2t

The derivative of -6x^2 with respect to t treats x as a constant, so it evaluates to 0.

Putting it all together, the derivative with respect to t is:
2x + 2t

Therefore, the mixed partial derivative is (d/dx)(d/dt)(x^2t + t^2x - 2x^3) = 2x + 2t.