Here are the dimensions of db^{[1]} and dZ^{[1]}:

db^{[1]} and b^{[1]} have dimensions n^{[1]} x 1, where n^{[1]} is the number of output neurons in layer 1.

dZ^{[1]} and Z^{[1]} are n^{[1]} x m, where m is the number of samples.

Note that all the “parameters”, meaning the W^{[l]} and b^{[l]} values, are independent of the number of samples. The forward propagation formulas work for any number of samples without changing the shapes of W and b.

Note that the derivation of all the back propagation formulas is beyond the scope of this course. Here’s a thread with links to other materials that cover the topic.