By taking the derivatives. 
But the point is you need to know vector and matrix calculus in order to derive the formulas shown here. The principles are the same as in univariate calculus, but the added dimensions make things a bit more complicated.
The high level answer is that Prof Ng has specifically designed these courses not to require any knowledge of calculus, even univariate calculus. So he does not show the actual derivation of most of the formulas. Here’s a thread with links to more information about all this and to some backgroound information about matrix calculus if you have some math background and want to dig deeper.
The good news is that you don’t need to know calculus, but the flip side of that is that you just need to take Prof Ng’s word for the formulas in that case.