Most ways to differentiate

[TheAspiringDoc]

Hey

This might take a bit of time, but I was wondering if someone could tell me a function/relation which can be differentiated in the most possible different ways using specialist techniques?

So far, the best of got is,

x = y^3

You can take the cube root of each side and then do it explicitly
You can find dx/dy and then invert it
You can do implicit

Thanks and good luck!

Edit: perhaps you could also swap the axes?
Or you could do each method by first principles also

[TheAspiringDoc]

Bump?

[TrueTears]

First you need to define precisely what you mean by "most possible different ways". Each of the "ways" you listed are actually equivalent to each other. The space of all functions that you consider is also too large, for example, some functions are not differentiable.

[TheAspiringDoc]

By “most possible different ways” I mean whatever can be differentiated use the greatest number of even slightly different VCE Specialist differentiation techniques. I know, they all essentially just stem from first principles, but I’m just talking about even small differences like doing it implicitly or rearranging it to do it explicitly, etc etc.

I know that some functions can’t be differentiated at certain points or at all, but I’m just looking for an example of a VCE-style function (or relation) which can be differentiated using several different strategies. By this I mean anything from completing the square to the product rule to the chain rule to implicit to converting it to parametric form (idk)

[TrueTears]

What I mean is, you need to define (in a mathematical sense) what you mean by "greatest number of techniques", otherwise your question is not well-defined. For instance, what is your definition of a "technique" and what is your definition of "different technique". To give an example of why such a definition is important, if I have a (differentiable) function, one "technique" of differentiating it is simply to +1-1 to the function and then differentiate it, another "different technique" could be to +2-2 to the function and then differentiate it, thus following such a process means there are infinitely many "techniques" to differentiate such a function.

For all differentiable real valued functions \(f: \mathbb{R} \rightarrow \mathbb{R}\), there is one and only one "way" to differentiate it, that is, from its definition. All other "ways" are equivalent. To make formal what I mean, let \(X\) be the set of all real valued (univariate) functions and let two functions be related to each other if they have the same derivative, then such a relation forms an equivalence class.

[TheAspiringDoc]

Ok.

Imagine you’re a student in an exam. You see “differentiate x=y³”
What do you do?

My first instinct would be to rearrange it to y=x^(1/3) and then to use the power rule of differentiation.
My friend might do it implicitly: 1 = 3y²dy/dx, so dy/dx = 1/(3y²)
Another friend might find find dx/dy and then invert that to get dy/dx
Another friend might find dx/dy and leave it at that.

But what might my other friends do? (Note: none of them will bother adding and subtracting 1 etc. - I’m talking about reasonable realistic useful efficient recognised functional practical convenient expected standard fruitful productive constructive efficacious valuable worthwhile alternative viable purposeful meaningful interesting non-pointless methods...)

What differentiation question on a realistic normal usual straightforward simple specialist test would potentially cause the greatest variety in the use of these ever so slightly different - even though fundamentally equivalent - techniques/methods/strategies?

[TrueTears]

Right, your initial question was a fundamentally different question because it is a ‘counting’ question and in order to count, you must define what differentiates the objects that you are counting. Again, to answer that question, one would still need a definition, eg, what do you define as a ‘non-pointless method’, you can not use descriptive English to define such an object, rather you must mathematically define it, otherwise my previous answer stands as there is an infinite number of ‘ways’ to differentiate.

Your current question is different; you are asking which differentiation technique to apply to differentiate a function (rather than count the number of ways). Such question depends on the function at hand and varies depending on what function you are looking at.

[TheAspiringDoc]

Your current question is different; you are asking which differentiation technique to apply to differentiate a function (rather than count the number of ways). Such question depends on the function at hand and varies depending on what function you are looking at

Hence my initial question

I was wondering if someone could tell me a function/relation which can be differentiated in the most possible different ways using specialist techniques?