Transport theorem, final integral

[Leb]

Homework Statement

Show that

\frac{d}{dt}\int \rho r^{2}\phi dr = \int \rho r^{2}\frac{d\phi}{dr} dr

Homework Equations

Fundamental theorem of calculus

The Attempt at a Solution

So I follow the derivation from the textbook and I think I get the rather sneaky rearrangement of the derivatives, but I do not see how
\int \rho r^{2}\frac{d\phi}{dt} dr = \int \rho r^{2}\left(\frac{\partial \phi}{\partial t}+v\frac{\partial \phi}{\partial r}\right)dr

Note: Integrals are evaluated from a to b, and v(x,t) = dx/dt (e.g. da/dt = v(a,t))

 

[Mindscrape]

What do the variables depend on? I don't see any problem with the line given a t and r dependence in phi. What subject is this in? Care to share some more problem details?

 

[Leb]

Sorry, I thought it was something standard. It is a mechanics course.

t-time, r - radius, \rho (r,t) is density, \phi (r,t) is an arbitrary differentiable function, a=a(t), b=b(t)

EDIT:

Nevermind, it is simply using the definition of the total derivative. That's all there is to it...