diff --git a/README.md b/README.md
index 7130e99..f9ad013 100644
--- a/README.md
+++ b/README.md
@@ -19,19 +19,19 @@ da = xr.DataArray(
 @eqx.filter_jit
 def some_function(data):
     neg_data = -1.0 * data
-    return neg_data * neg_data.coords["y"] # Multiply data by coords.
+    return neg_data * neg_data.coords["y"]  # Multiply data by coords.
 
 da = some_function(da)
 
 # Construct a xr.DataArray with dummy data (useful for tree manipulation).
-da_mask = jax.tree.map(lambda _: True, data)
+da_mask = jax.tree.map(lambda _: True, da)
 
-# Use jax.grad.
+# Take the gradient of a jitted function.
 @eqx.filter_jit
 def fn(data):
     return (data**2.0).sum().data
 
-grad = jax.grad(fn)(da)
+da_grad = jax.grad(fn)(da)
 
 # Convert to a custom XjDataArray, implemented as an equinox module.
 # (Useful for avoiding potentially weird xarray interactions with JAX).
@@ -39,7 +39,6 @@ xj_da = xj.from_xarray(da)
 
 # Convert back to a xr.DataArray.
 da = xj.to_xarray(xj_da)
-
 ```
 ## Installation
 ```bash
diff --git a/xarray_jax/tests/test_readme_examples.py b/xarray_jax/tests/test_readme_examples.py
index 1cb5cb3..54ee764 100644
--- a/xarray_jax/tests/test_readme_examples.py
+++ b/xarray_jax/tests/test_readme_examples.py
@@ -9,7 +9,12 @@
 """
 
 
-def test_dataarray_example():
+def test_main_example():
+    import jax.numpy as jnp
+    import xarray as xr
+    import xarray_jax as xj
+
+    # Construct a DataArray.
     da = xr.DataArray(
         xr.Variable(["x", "y"], jnp.ones((2, 3))),
         coords={"x": [1, 2], "y": [3, 4, 5]},
@@ -17,21 +22,29 @@ def test_dataarray_example():
         attrs={"attr1": "value1"},
     )
 
+    # Do some operations inside a JIT compiled function.
     @eqx.filter_jit
     def some_function(data):
-        neg_data = -1.0 * data  # Multiply data by -1.
+        neg_data = -1.0 * data
         return neg_data * neg_data.coords["y"]  # Multiply data by coords.
 
-    da_new = some_function(da)
+    da = some_function(da)
 
-    assert da_new.equals(-1.0 * da * da.coords["y"])
+    # Construct a xr.DataArray with dummy data (useful for tree manipulation).
+    da_mask = jax.tree.map(lambda _: True, da)
 
-    @eqx.filter_jit
+    # Take the gradient of a jitted function.    @eqx.filter_jit
     def fn(data):
         return (data**2.0).sum().data
 
-    grad = jax.grad(fn)(da)
-    assert grad.equals(2.0 * da)
+    da_grad = jax.grad(fn)(da)
+
+    # Convert to a custom XjDataArray, implemented as an equinox module.
+    # (Useful for avoiding potentially weird xarray interactions with JAX).
+    xj_da = xj.from_xarray(da)
+
+    # Convert back to a xr.DataArray.
+    da = xj.to_xarray(xj_da)
 
 
 def test_fail_example():
@@ -43,3 +56,12 @@ def test_fail_example():
     with pytest.raises(ValueError):
         # This will fail.
         var = var + 1
+
+    var = xr.Variable(dims=("x", "y"), data=jnp.ones((3, 3)))
+
+    # This will fail.
+    with pytest.raises(TypeError):
+        jnp.square(var)
+
+    # This will work.
+    xr.apply_ufunc(jnp.square, var)