Find insert/update/delete queries that reference a partitioned view
You can query from sql cache for long time:

SELECT cp.objtype AS ObjectType,
OBJECT_NAME(st.objectid,st.dbid) AS ObjectName,
cp.usecounts AS ExecutionCount,
st.TEXT AS QueryText,
qp.query_plan AS QueryPlan
FROM sys.dm_exec_cached_plans AS cp
CROSS APPLY sys.dm_exec_query_plan(cp.plan_handle) AS qp
CROSS APPLY sys.dm_exec_sql_text(cp.plan_handle) AS st


You can add a where condition for the view name and the insert/update keyword in st.text.
You can schedule a job and save te results in a table and then check if the table contains rows for update/insert statements against your view.