My last note went over differentiating and it turned out to be
However, I feel that there is one more problem for this trilogy of notes to acknowledge: the derivative of . We can utilise our generalisation from earlier to solve this:
But what is ? It is the same as
but can this be expressed more clearly? Maybe. Let us have another go at differentiating, but using a different method.
The equation is the same as . This is because . We can use implicit differentiation here to find the derivative of :
Note that so we can also write this as
Also remember that , so it can be written as
This must be the same as our first attempt on the derivative of , so we may be able to figure out what the infinite nested brackets could be:
So the derivative of is , but we also figured out that which is quite cool if i do say so myself.
I hope that you found this interesting!