y⁴ + x⁵ - 7x² - 5x^-1 = 0

We see how to derive this expression one part at a time. We just derive expressions as we come to them from left to right.

(In this example we could easily express it in terms of y, but this is intended as a relatively simple first example.)

Part A: Find the derivative with respect to x of: y⁴

To differentiate this expression, we regard y as a function of x and use the power rule.

Basics: Observe the following pattern of derivatives:

It follows that:

Part B: Find the derivative with respect to x of:

x⁵ - 7x² - 5x^-1

This is just ordinary differentiation:

Part C:

On the right hand side of our expression, the derivative of zero is zero. ie

Now, combining the results of parts A, B and C:

Next, solve for dy/dx and the required expression is: