While the geometric approach is useful, it may not be rigorous enough to establish the product rule. For a more rigorous approach, we can start from the definition of a limit (1) and work our way from there.
Let f(x) = u(x)v(x) then the derivative is given by
Here, we just substituted the function into the definition of the derivative. Looking at the final expression on the right, it seems there is no way to simplify this any further and it does not look like the product rule at all. We need to do some form of algebraic manipulation; we are going to add zero to the numerator.
Recall this trivial fact yet very useful here, A - A = 0, this is true if A is a number or a function.
So inserting expression (4) into the numerator of expression (3) will not change the expression since we are adding zero. We then get,
So far, we have arranged the terms, now on the last expression we have the limit of a sum, this is the same as the sum of the limits. Mathematically this reads
Both of these limits contain a product and we know the limit of a product is the same as the products of the limits, we get
The terms in bold are just the definitions of the derivative, and the terms in black simplify to the respective functions. This fact is true provided the function is continuous, and it will be true if the function is differentiable. So putting it all together, we get