hidden layer weights calculation I've been working on neural network for a while and I built simple network from scratch with python but before using TensorFlow, I would like to have a complete understanding of it.

Here is my question :

Lets say you have 3 layers you have 3 weights to update :

1) --> the weight between the outputlayer and the hiddenLayer2

2) -->  the weight between the hiddenLayer2 and the hiddenLayer1

3) -->  the weight between the hiddenLayer1 and the inputLayer

For the 1) the calculation is quite simple we got :

weight_3 += LEARNING_RATE * ((2*(target - output)) * sigmoid'(output) * layer2)


For the 2) the calculation is more complicated  and we got :

weight_2 += LEARNING_RATE * ((2*(target - output)) * sigmoid'(output) * weight_3) * sigmoid'(hiddenLayer2)


I need help for the 3rd part, I tried to calculate and find on internet but not a lot of people uses 2 hidden layer when they work from scratch.  

I also tried to resolve the chain rule but its too long and I can't resolve.

Does someone know the formula to get the weight between the hiddenLayer1 and the inputLayer ?

Thank you so much in advance
Here is a good example of how to implement forward and backward propagation, in numpy, but there should be similar functions in tf.