In the following example backpropagation will be performed on a single dense neurone that accepts 3 input variables.

The forward pass:

Where is the inputs to our neurones, is the weights of the connections between our inputs and our neurone, is the bias term of our neurone.

The backward pass:

This was just the partial derivatives with respect to but this can be applied to all the weights

This gives use the final gradients for our single neurones weights of

Now that these partial derivatives are written out explicitly they can also be written by keeping them as matrices

Note that in this case the bias term would also have a partial derivatives associated with it, even though it was omitted from this example.