Question

What is implicit differentiation?

Answer 1

I tried this hoping it is understandable!

Explanation:

You may remember the difference between an implicit and explicit form of a function:
Normally your function can be read explicitly as
`y="something of x"`
and derived immediately to find `(dy)/(dx)="something else of x"`;

For example `y=x^2-3x`
`(dy)/(dx)=2x-3`

Sometimes your function is more difficult to "extricate" and can be difficult to write as `y="something of x"`.
You can only write it as `y(x)+ax^n+...c=0`
As you can see the `y` is nested inside the other bits and it is in itself a function of `x`.

Consider for example a circle:
`x^2+y^2=4`
Here `y` is "difficult" to extract (ok, not impossible) and if you try to derive as it is, you have to remember to derive also `y`:
`2x+color(red)(2y(dy)/(dx))=0`
Rearranging:
`(dy)/(dx)=-(2x)/(2y)=-x/y`
As you can see this derivative contains again `y`; once you can figure out the form of `y` you could substitute it in and get the usual derivative containing only `x`.

In the example of the circle you can write it as (extracting `y`):
`y=+-sqrt(4-x^2)`
So that gives you:
`(dy)/(dx)=-x/(color(red)(+-sqrt(4-x^2))`

It isn't always possible to extract `y` and you leave the derivative as it is, with `y` in it.

To test this result try to derive the explicit `y=+-sqrt(4-x^2)` form and see if it checks with the one we found implicitly.