diff --git a/cdfm/models/differentials.py b/cdfm/models/differentials.py
index 6a2ed96..81c5348 100644
--- a/cdfm/models/differentials.py
+++ b/cdfm/models/differentials.py
@@ -3,6 +3,7 @@
 """
 from typing import List
 import numpy as np
+from ..config import DTYPE
 
 
 def Iec_ve(cinds: List[int], Vc: np.ndarray) -> np.ndarray:
@@ -29,6 +30,55 @@ def Iec_vc(eind: int, Ve: np.ndarray) -> np.ndarray:
     """
     return Ve[eind]
 
+def p_Iec_u(eind: int,
+            cinds: int,
+            Ve: np.ndarray,
+            Vc: np.ndarray,
+            d: np.ndarray,
+            ps: np.ndarray) -> np.ndarray:
+    """Partially differentiate p_Iec with u.
+
+    Parameters:
+        eind: an index of the target entity.
+        cinds: indices of competitors.
+        Ve: a matrix of latent vectors of entities. shape (p, k).
+        Vc: a matrix of latent vectors of competitors. shape (p, k).
+        d: a proximity factors. shape (|cinds|, #factors).
+        ps: a probabilities of interaction occurrence.
+
+    Returns:
+        res: shape (#factors, ).
+    """
+    partial_sigmoid = np.einsum('i,ij->ij', np.multiply(ps, (DTYPE(1.) - ps)), -d)
+    strengths = np.dot(Vc[cinds], Ve[eind])
+    return np.einsum('ij,i->j', partial_sigmoid, strengths)
+
+def p_Iec_ve(cinds: List[int], Vc: np.ndarray, ps: np.ndarray) -> np.ndarray:
+    """Partially differentiate P_Iec with ve.
+
+    Parameters:
+        cinds: indices of competitors.
+        Vc: a matrix of latent vectors of competitors. shape (p, k).
+        ps: a probabilities of interaction occurrence.
+
+    Returns:
+        res: shape (k, ).
+    """
+    return np.einsum('ij,i->j', Vc.take(cinds, axis=0), ps)
+
+def p_Iec_vc(eind: int, Ve: np.ndarray, p: DTYPE) -> np.ndarray:
+    """Partially differentiate P_Iec with ve.
+
+    Parameters:
+        eind: an index of the target entity.
+        Ve: a matrix of latent vectors of entities. shape (p, k).
+        p: a probability of interaction occurrence.
+
+    Returns:
+        res: shape (k, ).
+    """
+    return np.multiply(p, Ve[eind])
+
 def Ief_ve(x: np.ndarray, Vf: np.ndarray) -> np.ndarray:
     """Partially differentiate Ief with ve.
 
diff --git a/tests/models/test_differentials.py b/tests/models/test_differentials.py
index 90835bf..00cca9b 100644
--- a/tests/models/test_differentials.py
+++ b/tests/models/test_differentials.py
@@ -1,5 +1,7 @@
 # -*- coding: utf-8 -*-
 """Testing differential functions.
+
+Checking result, type, dtype, and shape.
 """
 import numpy as np
 from cdfm.config import DTYPE
@@ -38,15 +40,52 @@ def test_Iec_ve(self):
         cinds = [1, 2]
         res = Diff.Iec_ve(cinds, self.Vc)
         expected = np.array([11.0, 13.0, 15.0], dtype=DTYPE)
-        assert (res == expected).all()
+        assert np.allclose(res, expected)
         assert isinstance(res, np.ndarray)
         assert res.dtype == DTYPE
+        assert res.shape == (3, )
 
     def test_Iec_vc(self):
+        """eind: 0
+        """
         eind = 0
         res = Diff.Iec_vc(eind, self.Ve)
         expected = np.array([1.0, 2.0, 3.0], dtype=DTYPE)
-        assert (res == expected).all()
+        assert np.allclose(res, expected)
+        assert isinstance(res, np.ndarray)
+        assert res.dtype == DTYPE
+        assert res.shape == (3, )
+
+    def test_p_Iec_u(self):
+        eind = 0
+        cinds = [1, 2]
+        d = np.array([[1, 4], [2, 3]], dtype=DTYPE)
+        ps = np.array([0.3, 0.7], dtype=DTYPE)
+        res = Diff.p_Iec_u(eind, cinds, self.Ve, self.Vc, d, ps)
+        expected = np.array([-27.72, -58.38], dtype=DTYPE)
+        assert np.allclose(res, expected)
+        assert isinstance(res, np.ndarray)
+        assert res.dtype == DTYPE
+
+    def test_p_Iec_ve(self):
+        """eind: 0, cinds: 1, 2
+        """
+        cinds = [1, 2]
+        ps = np.array([0.3, 0.7], dtype=DTYPE)
+        res = Diff.p_Iec_ve(cinds, self.Vc, ps)
+        expected = np.array([6.1, 7.1, 8.1], dtype=DTYPE)
+        assert np.allclose(res, expected)
+        assert isinstance(res, np.ndarray)
+        assert res.dtype == DTYPE
+
+    def test_p_Iec_vc(self):
+        """eind: 0, (cind: 1), p: 0.3 (between 0 and 1)
+        """
+        eind = 0
+        p = DTYPE(0.3)
+        res = Diff.p_Iec_vc(eind, self.Ve, p)
+        expected = np.array([0.3, 0.6, 0.9])
+        assert np.allclose(res, expected)
         assert isinstance(res, np.ndarray)
         assert res.dtype == DTYPE
 
@@ -54,7 +93,7 @@ def test_Ief_ve(self):
         x = np.ones(4, dtype=DTYPE)
         res = Diff.Ief_ve(x, self.Vf)
         expected = np.array([22, 26, 30], dtype=DTYPE)
-        assert (res == expected).all()
+        assert np.allclose(res, expected)
         assert isinstance(res, np.ndarray)
         assert res.dtype == DTYPE
 
@@ -68,7 +107,7 @@ def test_Ief_vf(self):
             [1.0, 2.0, 3.0],
             [1.0, 2.0, 3.0]
         ], dtype=DTYPE)
-        assert (res == expected).all()
+        assert np.allclose(res, expected)
         assert isinstance(res, np.ndarray)
         assert res.dtype == DTYPE
 
@@ -81,6 +120,6 @@ def test_Iff_vf(self):
             [-141.0, -156., -171.0],
             [-56.0, -64.0, -72.0],
         ], dtype=DTYPE)  # culculated by hand
-        assert (res == expected).all()
+        assert np.allclose(res, expected)
         assert isinstance(res, np.ndarray)
         assert res.dtype == DTYPE