Suppose someone does research on a phenomenon and builds the model
$y=a+bx$.
The scientist then states the "Theorem" that, given the particular assumptions leading to the model, $x$ affects $y$ positively.
Finally, as "Proof" the scientist notes that differentiating the model with respect to $x$ gives $b$, which is positive.
Although this is a very simple example, can we say that we are indeed dealing with a theorem and a proof? If so, what kind of proof are we dealing with?
Edits:
A comment below mentions that the example given is "a more or less trivial result of calculus". But imagine if the model was much more complicated, and an effect of a variable was again proposed. Then, are things like taking derivatives and finding roots still considered none-proof?
Another comment argues that the example is merely a (partial) confirmation of a system. What is the difference between a confirmation and a proof?
Rationale:
The reason I ask this question, and why I hope it will not be closed, is that I have read papers (practical in scope, yet mathematical), where the author provides a theorem, and then states a proof along the lines: "The result is obtained by differentiation the equation with respect to x, setting this derivative to zero, and noting it is zero only when (insert some conditions)"
If this kind of written argumentation is a misuse of terms, what should the author instead have written?