Question about learning with gradient descent • /r/MachineLearning

Once one has computed all derivatives of the cost with respect to weights and biases / errors, what, exactly, does one do? How does one update the weights/biases? I can think of a simple way of doing it for a net with no hidden layer, but I have no idea how one would do it on one with more layers, since changing any weight or bias will change the value of the error of neurons in earlier and later layers - hence my question.