differentiating diffusion pde under the integral sign

Question

I got little bit confused about how we differentiate the diffusion PDE: -Let u(x, t) be the concentration (mass per unit length) of the dye at position x of the pipe at time t. -In the section of pipe from x0 to x1 , the mass of dye is: $$M(t)= \int_{x_0}^{x_1}u(x,t)dx\Longrightarrow \frac{dM}{dt}=\int_{x_0}^{x1}u_t(x,t)dx $$ and differentiating with respect to $x_1 $ We get $u_t$ could anyone help me out ? Thanks