diff --git a/app/vmselect/promql/exec_test.go b/app/vmselect/promql/exec_test.go
index 7ec9f90427..789726f18b 100644
--- a/app/vmselect/promql/exec_test.go
+++ b/app/vmselect/promql/exec_test.go
@@ -6419,9 +6419,9 @@ func TestExecSuccess(t *testing.T) {
 		resultExpected := []netstorage.Result{r}
 		f(q, resultExpected)
 	})
-	t.Run(`deriv(1)`, func(t *testing.T) {
+	t.Run(`deriv(N)`, func(t *testing.T) {
 		t.Parallel()
-		q := `deriv(1)`
+		q := `deriv(1000)`
 		r := netstorage.Result{
 			MetricName: metricNameExpected,
 			Values:     []float64{0, 0, 0, 0, 0, 0},
diff --git a/app/vmselect/promql/rollup.go b/app/vmselect/promql/rollup.go
index f650f9bc48..2ef33f6e44 100644
--- a/app/vmselect/promql/rollup.go
+++ b/app/vmselect/promql/rollup.go
@@ -850,7 +850,7 @@ func linearRegression(rfa *rollupFuncArg) (float64, float64) {
 	if n == 0 {
 		return nan, nan
 	}
-	if n == 1 {
+	if areConstValues(values) {
 		return values[0], 0
 	}
 
@@ -877,6 +877,20 @@ func linearRegression(rfa *rollupFuncArg) (float64, float64) {
 	return v, k
 }
 
+func areConstValues(values []float64) bool {
+	if len(values) <= 1 {
+		return true
+	}
+	vPrev := values[0]
+	for _, v := range values[1:] {
+		if v != vPrev {
+			return false
+		}
+		vPrev = v
+	}
+	return true
+}
+
 func newRollupDurationOverTime(args []interface{}) (rollupFunc, error) {
 	if err := expectRollupArgsNum(args, 2); err != nil {
 		return nil, err
diff --git a/docs/CHANGELOG.md b/docs/CHANGELOG.md
index e25bfc5b8d..8c9ec1ae0f 100644
--- a/docs/CHANGELOG.md
+++ b/docs/CHANGELOG.md
@@ -15,6 +15,7 @@ sort: 15
 * BUGFIX: vmauth: properly take into account the value passed to `-maxIdleConnsPerBackend` command-line flag. See [this issue](https://github.com/VictoriaMetrics/VictoriaMetrics/issues/1300).
 * BUGFIX: vmagent: fix [reading data from Kafka](https://docs.victoriametrics.com/vmagent.html#reading-metrics-from-kafka).
 * BUGFIX: vmalert: fix [replay mode](https://docs.victoriametrics.com/vmalert.html#rules-backfilling) in enterprise version.
+* BUGFIX: consistently return zero from [deriv()](https://docs.victoriametrics.com/MetricsQL.html#deriv) function applied to a constant time series. Previously it could return small non-zero values in this case.
 
 
 ## [v1.69.0](https://github.com/VictoriaMetrics/VictoriaMetrics/releases/tag/v1.69.0)
diff --git a/docs/vmauth.md b/docs/vmauth.md
index 827e678e90..43a2605672 100644
--- a/docs/vmauth.md
+++ b/docs/vmauth.md
@@ -47,6 +47,7 @@ Each `url_prefix` in the [-auth.config](#auth-config) may contain either a singl
 users:
   # Requests with the 'Authorization: Bearer XXXX' header are proxied to http://localhost:8428 .
   # For example, http://vmauth:8427/api/v1/query is proxied to http://localhost:8428/api/v1/query
+  # Requests with the Basic Auth username=XXXX are proxied to http://localhost:8428 as well.
 - bearer_token: "XXXX"
   url_prefix: "http://localhost:8428"