deriveChild and signing transactions using Bit-Wasp

I’m generating multiple children using one extended key. I have no problem generating public addresses:

$master = HierarchicalKeyFactory::fromExtended(myKey);
$key = $master->deriveChild(1);
$publicAddress = $key->getPublicKey()->getAddress()->getAddress();

In order to sign a transaction, I need the public address’ private key. How do I generate / extract the private key of a derived child? I’m assuming I’m not using the extended key for that.