The root cause of the CWE-476 (NULL Pointer Dereference) vulnerability in the given code slice is:

The code attempts to access the 'clientlist_mutex' member of the 'dev' structure without first checking if 'dev' is a valid (non-NULL) pointer. If 'client->dev' is NULL, dereferencing it on line 6 will lead to a NULL pointer dereference, potentially causing a program crash or undefined behavior.

To prevent this vulnerability, the code should include a NULL check for 'dev' before attempting to lock the mutex. A corrected version might look like:

```c
void drm_client_register(struct drm_client_dev *client)
{
    struct drm_device *dev = client->dev;
    int ret;

    if (dev == NULL) {
        // Handle the error appropriately, e.g., return or log an error
        return;
    }

    mutex_lock(&dev->clientlist_mutex);
    // Rest of the function...
}
```

This check ensures that 'dev' is not NULL before attempting to access its members, preventing the potential NULL pointer dereference.