This is known as the Hotelling's Lemma. To do this, we assume that 𝜋,𝑘,𝑙 are differentiable wrt 𝑝,𝑣,𝑤. What's the guarantee that such is the case with these variables?
If you have only generic functions as $\pi(p,v,w),k(p,v,w),l(p,v,w)$ , and you know nothing about them, you can't prove in any way that they are differentiable with respect $p, v$ and $w$, but you have to assume that these derivatives exist.
The case is different if you have specific functions. Here you can resort to the definition of partial derivatives or to theorems about the derivatives.
Can we prove the differentiability using possibly the first principles or in some other way?
The link to Wikipedia you quoted gives the definitions of differentiability of a function, both for function from $\mathbb{R}$ to $\mathbb{R}$ and functions of several real variables (to $\mathbb{R^n}$).
Here, we have to make a fundamental distinction, between functions of one real variable and functions of several real variables.
For a function from $\mathbb{R}$ to $\mathbb{R}$, differentiability means the existence of the first derivative, the definition of which is given by Wikipedia, that is the usual definition as the limit of the incremental ratio, and a function is differentiable if this limit exists (finite).
So, to check if a function is differentiable, you can use, as you know from basic mathematical analysis, the definition, calculating that limit for your specific function, or proving that it doesn't existe or is not finite, or resorting to theorems about some specific function being differentiable, or to theorems as 'the sum of two differentiable functions is differentiable', or that the composition of two differentiable functions is differentiable, and so on.
Very different is the case of functions of several variables.
In the article of Wikipedia we have the definition of differentiability of a function of several variables, as the existence of the differential of the function (also called total differential or total derivative). The differential of a function is the linear form (also called differential form) defined by Wikipedia through a particular limit (this is too long to report here).
But differentiability in this sense is not the same concept as the existence of the partial derivatives, as your derivatives of the functions $\pi(p,v,w),k(p,v,w),l(p,v,w)$. You, here, in the Hotelling Lemma, are considering not the differentiability in the sense of the existence of the differential, but only of the partial derivatives.
In the case of several functions there is this, one can say crucial, distinction.
The existence of partial derivative and the existence of the differential are distinct, even if correlated, concepts. For example, differentiability implies the existence of the derivative in all directions (not only partial derivatives), but the viceversa doesn’t hold.
The existence of the partial derivatives is not a notion equivalent to the differentiability of a function in one single variable, as, for instance, their existence in a point doesn't guarantee the continuity of the function in that point.
This is, instead, guaranteed by the existence of the differential, so that the actual generalization to several variables of the concept of differentiability in one variable is the differential.
One can see this from a geometrical point of view: while the existence of the derivative in point of a function of one variable guarantees the existence of the tangent line in that point, the existence of the differential in a point guarantees the existence of the tangent plane in that point.
Practical conclusions: The differential is of course a very important concept from a theoretical point of view, but if you have to check the existence of partial derivatives of a function of several variables, you can forget the definition of differential (differentiability in the sense described by Wikipedia) and look at the partial derivatives only.
As the partial derivatives are actually, by definition, derivatives of a function of one variable, you can apply to partial derivatives the same methods as for functions of one real variable.
I hope this short synthesis of a complex issue can be useful to you to contextualize your problem and find your way in this subject.
