*y*^{4}
+ *x*^{5} −
7*x*^{2} −
5*x*^{-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:
***y*^{4}

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

**Basics:** Observe the following pattern of
derivatives:

`d/(dx)y=(dy)/(dx)`

`d/(dx)y^2=2y(dy)/(dx)`

`d/(dx)y^3=3y^2(dy)/(dx)`

It follows that:

`d/(dx)y^4=4y^3(dy)/(dx)`

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

x^{5}− 7x^{2}− 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

`d/(dx)(0)=0`

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

`4y^3(dy)/(dx)+5x^4-14x+5x^-2=0`

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

`(dy)/(dx)=(-5x^4+14x-5x^-2)/(4y^3`