diff --git a/CHANGELOG.md b/CHANGELOG.md
index 1c341d8756ee..c417f5c17f6b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -10,6 +10,9 @@ PLEASE REMEMBER TO CHANGE THE '..main' WITH AN ACTUAL TAG in GITHUB LINK.
 
 ## jax 0.3.17 (Unreleased)
 * [GitHub commits](https://github.com/google/jax/compare/jax-v0.3.16...main).
+* Bugs
+  * Fix corner case issue in gradient of `lax.pow` with an exponent of zero
+    ({jax-issue}`12041`)
 * Breaking changes
   * {func}`jax.checkpoint`, also known as {func}`jax.remat`, no longer supports
     the `concrete` option, following the previous version's deprecation; see
diff --git a/jax/_src/lax/lax.py b/jax/_src/lax/lax.py
index 6d9d1af78d08..eb618a59fc3c 100644
--- a/jax/_src/lax/lax.py
+++ b/jax/_src/lax/lax.py
@@ -2004,8 +2004,7 @@ def _abs_jvp_rule(g, ans, x):
 pow_p = standard_naryop([_float | _complex, _float | _complex], 'pow')
 
 def _pow_jvp_lhs(g, ans, x, y):
-  jac = mul(y, pow(x, select(eq(y, _zeros(y)), _ones(y), sub(y, _ones(y)))))
-  return mul(g, jac)
+  return mul(g, mul(y, pow(x, sub(y, _ones(y)))))
 
 def _pow_jvp_rhs(g, ans, x, y):
   return mul(g, mul(log(_replace_zero(x)), ans))
diff --git a/tests/lax_autodiff_test.py b/tests/lax_autodiff_test.py
index e9c7bb6b6e26..f62e6ae6f8b2 100644
--- a/tests/lax_autodiff_test.py
+++ b/tests/lax_autodiff_test.py
@@ -529,6 +529,26 @@ def testReverseGrad(self):
     check_grads(rev, (np.array([[6., 5., 4.], [3., 2., 1.]]),), 2,
                 rtol={np.float32: 3e-3})
 
+  def testPowSecondDerivative(self):
+    # https://github.com/google/jax/issues/12033
+    x, y = 4.0, 0.0
+    expected = ((0.0, 1/x), (1/x, np.log(x) ** 2))
+
+    with self.subTest("jacfwd"):
+      result_fwd = jax.jacfwd(jax.jacfwd(lax.pow, (0, 1)), (0, 1))(x, y)
+      self.assertAllClose(result_fwd, expected)
+
+    with self.subTest("jacrev"):
+      result_rev = jax.jacrev(jax.jacrev(lax.pow, (0, 1)), (0, 1))(x, y)
+      self.assertAllClose(result_rev, expected)
+
+    with self.subTest("zero to the zero"):
+      result = jax.grad(lax.pow)(0.0, 0.0)
+      # TODO(jakevdp) special-case zero in a way that doesn't break other cases
+      # See https://github.com/google/jax/pull/12041#issuecomment-1222766191
+      # self.assertEqual(result, 0.0)
+      self.assertAllClose(result, np.nan)
+
   @parameterized.named_parameters(jtu.cases_from_list(
       {"testcase_name": "_predshape={}_argshapes={}".format(
           jtu.format_shape_dtype_string(pred_shape, np.bool_),