Dot product and elementwise multiply are completely different operations, of course. It depends on how the math works out. Some operations are “elementwise” in forward propagation, e.g. the application of the activation functions. So when you apply the Chain Rule to that operation, you will end up with elementwise multiplies.
Note that Prof Ng has specifically designed these courses not to require knowledge of calculus. That means he cannot show us the actual derivations of back propagation: they are literally beyond the scope of these courses. He just gives us the formulas and we take his word for it. If you have the math background to understand vector and matrix calculus, here’s a thread with links to resources that cover the derivation of back propagation.