So I am trying to derive the heat equation and I have the following equation
$$-\frac{1}{\Delta x}\int_{x}^{x+\Delta x } \frac{d}{dt} \phi(x,t)dx = u_t(x,t)$$ $$\phi(x,t)=-K\frac{du}{dx}$$ where K is a constant and $u_{xx}$ is $\frac{d}{dx}\phi(x,t)$
Now my issue is understanding what happens when you take the limit as $\Delta x$ approaches zero. Does the fundamental theorem say that the left-hand side becomes $\frac{d}{dx} \phi(x,t)$ or what? I'm very confused as to how to get to the $u_{xx}$ form of the left-hand side.
Here is just a hint. Consider $$g(x_0) = \int_{-\infty}^{x_0}f(x)dx$$ Now we can verify that we will get: $$\frac{g(x_0+\Delta x) - g(x_0)}{\Delta x} = \frac{1}{\Delta x}\int_{x_0}^{x_0+\Delta x}f(x)dx$$
Do you recognize the expression to the left? What happens when you take the considered limit on it?