The first fundamental theorem of calculus

[Mr Davis 97]

Say I have the statement $\int \frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d}x = y$. How does the fundamental theorem of calculus make this necessarily true? When I see the formal statement of the theorem, it is usually in terms of a definite integral such as $F(x) = \int_{a}^{x}f(t)dt$. How does the later apply to the former if the former is an antiderivative?

 

[allisson]

BUT Y= F(X)
SO, dy/dx=f(x)
la segunda es integral definida

 

[Raghav Gupta]

Say I have the statement $\int \frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d}x = y$. How does the fundamental theorem of calculus make this necessarily true? When I see the formal statement of the theorem, it is usually in terms of a definite integral such as $F(x) = \int_{a}^{x}f(t)dt$. How does the later apply to the former if the former is an antiderivative?

Do you know the Leibniz rule?