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 the function in terms of y only, 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:

`d/(dx)(x^5-7x^2-5x^-1)` `=5x^4-14x+5x^-2`

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: