10.9 Implicit differentiation
This is "just" an application of the chain rule!
This depends very much on what is. Usually, however, when is written, what is really meant is . Therefore, here, we really have
If we define a new function equal to , we then have
We can then use the chain rule 1313 13 Explained in the section above.
As we know that , we can then write the derivative of with respect to as
10.9.1 Optimisation using implicit differentiation
The point exists in the x-y plane. The circle is defined as
Which point on is the furthest from ?
Solution: The main thing here is to clearly define the problem in terms of algebraic relations (i.e. equations) which make it easy to solve the problem using differentiation.
What we’re trying to maximise is the distance of some unknown point - which we’ll call - from the point . We’re not after any old point, though! For our points and we also require that (as they must lie on the circle).
We can define a function which outputs the distance between and any two points as
We then know that this function’s turning points (and thus the minima and maxima) will be when
We can find the derivative using implicit differentiation
and we want to know when this is equal to zero, which means that we can write that
What we’d usually try to do here is substitute either for or for into the function which gives us the relationship between and (in this case, C, as defined in Equation 10.58). Currently, though, this isn’t possible as there’s a throwing a spanner in the works. If we differentiate Equation 10.58, however, we can then express in terms of and , and thus eliminate it from the equation.
Returning to Equation 10.64, we can now elimate .
We can now subsitute this equation into (aka Equation 10.58), and find the values we’ve been after all this time.
Because there are two cases: in the first
In the second case instead
Plugging the two possible values of and into (hello again distance function - last seen in Equation 10.59), we get (using a handy pocket calculator) that is the further point from , and thus the furthest point is