ANN - Backpropagation with multiple output neurons
Backpropagation works for feed-forward ANNs with multiple output neurons.

For an output neuron $z_r$ in an ANN, the delta function is

$$
\delta_r = \frac{\partial f_r(a_r)}{\partial a_r} e_r\qquad,
$$
where $f_r(a_r)$ is the activation function of $z_r$, $a_r$ is the sum of weighted inputs of $z_r$ and $e_r$ is the error of $z_r$, $e_r$ = $z_r$ - $y_r$ ($y_r$ is the target value).

for a hidden neuron $z_k$ the delta function is the sum of weighted deltas of the neurons in the next layer (in forward direction) of $z_k$. So, for a neuron in the layer next to the output layer, the delta function is:

$$
\delta_k = \frac{\partial f_k(a_k)}{\partial a_k} \sum_{r\in post(k)}\delta_r w_{kr}\qquad,
$$

where $w_{kr}$ is the weight between neurons $z_k$ and $z_r$. I suggest to have a look at the Wikipedia article about Backpropagation, especially the subsection 'finding the derivative of the error'. Following the link at the end of this section (https://www4.rgu.ac.uk/files/chapter3%20-%20bp.pdf), the first five pages or so might be helpful for a better understanding.