diff --git a/src/Monotonic.cpp b/src/Monotonic.cpp
index aa4a049b0b64..e8dc33afcc81 100644
--- a/src/Monotonic.cpp
+++ b/src/Monotonic.cpp
@@ -139,11 +139,12 @@ class DerivativeBounds : public IRVisitor {
             ConstantInterval rb = result;
 
             // This is essentially the product rule: a*rb + b*ra
-            // but only implemented for the case where a or b is constant.
-            if (auto b = as_const_int_or_uint(op->b)) {
-                result = ra * (*b);
-            } else if (auto a = as_const_int_or_uint(op->a)) {
-                result = rb * (*a);
+            // but only implemented for the case where a or b doesn't
+            // depend on the var.
+            if (rb.is_single_point(0)) {
+                result = ra * constant_integer_bounds(op->b);
+            } else if (ra.is_single_point(0)) {
+                result = rb * constant_integer_bounds(op->a);
             } else {
                 result = ConstantInterval::everything();
             }
@@ -179,6 +180,34 @@ class DerivativeBounds : public IRVisitor {
                 }
                 return;
             }
+            op->b.accept(this);
+            if (result.is_single_point(0)) {
+                // The denominator doesn't depend on the var. If it's known
+                // to be nonzero, the quotient moves in the same direction as
+                // the numerator, or the opposite one, by at most as much.
+                ConstantInterval rb = constant_integer_bounds(op->b);
+                bool negative = rb < 0;
+                if (negative) {
+                    rb = -rb;
+                }
+                if (rb > 0) {
+                    op->a.accept(this);
+                    ConstantInterval ra = result;
+                    result = ConstantInterval::everything();
+                    if (ra.min_defined) {
+                        result.min_defined = true;
+                        result.min = std::min<int64_t>(div_imp(ra.min, rb.min), 0);
+                    }
+                    if (ra.max_defined && ra.max != INT64_MIN) {
+                        result.max_defined = true;
+                        result.max = std::max<int64_t>(div_imp(ra.max - 1, rb.min) + 1, 0);
+                    }
+                    if (negative) {
+                        result = -result;
+                    }
+                    return;
+                }
+            }
         }
         result = ConstantInterval::everything();
     }
diff --git a/test/correctness/is_monotonic.cpp b/test/correctness/is_monotonic.cpp
index 51385e656b72..fdcdc2b6ae0b 100644
--- a/test/correctness/is_monotonic.cpp
+++ b/test/correctness/is_monotonic.cpp
@@ -56,7 +56,15 @@ int main() {
     check_decreasing(x <= y);
     check_decreasing(x < y);
     check_unknown(x * y);
-
+    check_unknown(x / y);
+
+    // Scaling by something of known sign that doesn't depend on x.
+    check_increasing(x * max(y, 1));
+    check_increasing(max(y, 0) * x);
+    check_increasing(x / max(y, 1));
+    check_decreasing(x * min(y, -1));
+    check_decreasing(x / min(y, -1));
+    check_decreasing((4 - x) / max(y, 1));
     // Not constant despite having constant args, because there's a side-effect.
     check_unknown(Call::make(Int(32), "foo", {Expr(3)}, Call::Extern));
 
diff --git a/test/correctness/sliding_window.cpp b/test/correctness/sliding_window.cpp
index e7b0d86a433a..ac54a9f56f44 100644
--- a/test/correctness/sliding_window.cpp
+++ b/test/correctness/sliding_window.cpp
@@ -439,6 +439,25 @@ int main(int argc, char **argv) {
         }
     }
 
+    {
+        // Sliding over a division by a param of known sign.
+        count = 0;
+        Param<int> k;
+        Func f, g;
+        f(x) = call_counter(x, 0);
+        Expr i = x / max(k, 1);
+        g(x) = f(i) + f(i + 1);
+        f.store_root().compute_at(g, x);
+
+        k.set(3);
+        g.realize({100});
+        // f spans x in [0, 34], so 35 calls when slid.
+        if (count != 35) {
+            printf("f was called %d times instead of %d times\n", count, 35);
+            return 1;
+        }
+    }
+
     printf("Success!\n");
     return 0;
 }