Diagnose database at database level

Here is the script give us I/O statistics in each data/log file

SELECT DB_NAME(fs.database_id) AS [Database Name], mf.physical_name,
io_stall_read_ms, num_of_reads,
CAST(io_stall_read_ms/(1.0 + num_of_reads) AS NUMERIC(10,1)) AS
[avg_read_stall_ms],io_stall_write_ms,
num_of_writes,CAST(io_stall_write_ms/(1.0+num_of_writes) AS NUMERIC(10,1)) AS
[avg_write_stall_ms],
io_stall_read_ms + io_stall_write_ms AS [io_stalls], num_of_reads + num_of_writes
AS [total_io],
CAST((io_stall_read_ms + io_stall_write_ms)/(1.0 + num_of_reads + num_of_writes) AS
NUMERIC(10,1))
AS [avg_io_stall_ms]
FROM sys.dm_io_virtual_file_stats(null,null) AS fs
INNER JOIN sys.master_files AS mf WITH (NOLOCK)
ON fs.database_id = mf.database_id
AND fs.[file_id] = mf.[file_id]
ORDER BY avg_io_stall_ms DESC OPTION (RECOMPILE);

Here is the script show us the amount buffer is used for each database.

-- Get total buffer usage by database for current instance
SELECT DB_NAME(database_id) AS [Database Name],
COUNT(*) * 8/1024.0 AS [Cached Size (MB)]
FROM sys.dm_os_buffer_descriptors WITH (NOLOCK)
WHERE database_id > 4 -- system databases
AND database_id <> 32767 -- ResourceDB
GROUP BY DB_NAME(database_id)
ORDER BY [Cached Size (MB)] DESC OPTION (RECOMPILE);
-- Tells you how much memory (in the buffer pool)
-- is being used by each database on the instance

Refer: ebook SQL Internall and troubleshooting