How i can write codes to train a backpropagation network that can run on MATLAB without the neural network toolbox

I did some modelling to predict data using the neural network toolbox and the results i obtained were very satisfactory. However i need assistance with the codes i can use to create a back propagation neural network that can run on MATLAB without the neural network toolbox. I believe the codes will help me understand how the cost function is derived, how the gradients are computed, the regression parameters for every neuron, and also vary the regression parameters to see it's effect on the cost function which is more valuable from a research perspective.

Thanks