The derivative formula of fraction is an important concept in calculus, which describes the tangent slope of a function at a certain point. The formula is: (f(x)/g(x)' = f' (x) * g (x)-f (x) * g' (x), where f (x) and g (x) are two functions and' stands for derivative.

There are many ways to prove this formula, one of which is to use the concept of limit. First, we need to define a new function h(x)=f(x)/g(x). Then, we can calculate the derivative of h(x) at X. According to the definition of derivative, H' (x) = lim (δ x->; 0)[h(x+δx)-h(x)]/δx .

Next, we need to decompose h'(x) into two parts. The first part is f'(x)*g(x), and the second part is f(x)*g'(x). To prove this, we can use the chain rule. Chain rule is a basic rule in calculus, which describes how the derivative of a compound function is influenced by the derivative of its components.

According to the chain rule, h' (x) = h (x)' * g (x)+f (x) * h' (x) * g' (x). We can see that the first part of h'(x) is f'(x)*g(x), and the second part is f(x)*g'(x). So we get h'(x)=f'(x)*g(x)-f(x)*g'(x).

This is the proof of fractional derivative formula. This formula has applications in many fields of mathematics and science, including physics, engineering and economics. By understanding and mastering this formula, we can better understand and analyze various complex functional relationships.