when to use dot product and when to use the common product In neural networks?

Question

I wanted to know when to use dot product and when to not , I also don't know when we must transpose an array and why should we , could someone help me to understand this ? If you could give me the derivative of cost function with respect to each Weigh (W1, W2, W3) using notations bellow to compare it with the results I found I'll appreciate that !