y = 5x² - 4/x³

1/x can be written as x^-1, which means our equation can also be written as y = 5x² - 4x^-3.

dy/dx means that we need to differentiate y in terms of x.

To differentiate an equation, we need to multiply the coefficient (the number before the x) by its power (the smaller number above it), and then subtract 1 from the power. This must be done for all parts of the equation.

We can split up the equation and do the working bit by bit, so first lets look at "5x²":

Multiplying the coefficient by the power, we get 5 X 2 = 10, and then 2 - 1 = 1, which means the differential of 5x² is 10x¹, and the ^1 can be dropped to get 10x.

Looking at the second part "-4x^-3":

-4 X -3 = 12, and -3 - 1 = -4, so the differential of -4x^-3 is 12x^-4, which can also be written as 12/x⁴ (reversing the rule we used earlier).

So putting the two parts back together, we get dy/dx = 10x + 12/x⁴.

It is also important to note that the question specified x is not equal to 0, and this is due to the fact that division by 0 can have a significant effect on an equation, with any number divided by 0 equalling infinity, a very difficult number to quantify or use.