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:
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: