diff --git a/src/kernel_gravity.h b/src/kernel_gravity.h
index 5f52c7fe857bfc042404d972144d1da76139841a..1c2eb452a3e43f414758d589b79e34c84040a4a0 100644
--- a/src/kernel_gravity.h
+++ b/src/kernel_gravity.h
@@ -209,10 +209,10 @@ __attribute__((const)) INLINE static float D_soft_5(const float u,
   error("Invalid choice of softening kernel shape");
 #endif
 
-  /* (phi'(u)/u)'/u = 105u^3 - 360u^2 + 420u - 168 */
-  float phi = 105.f * u - 360.f;
-  phi = phi * u + 420.f;
-  phi = phi * u - 168.f;
+  /* (phi'(u)/u)'/u = 35u^3 - 120u^2 + 140u - 56 */
+  float phi = 35.f * u - 120.f;
+  phi = phi * u + 140.f;
+  phi = phi * u - 56.f;
 
   return phi;
 }
@@ -223,7 +223,9 @@ __attribute__((const)) INLINE static float D_soft_7(const float u,
   error("Invalid choice of softening kernel shape");
 #endif
 
-  return 315.f * u - 720.f + 420.f * u_inv;
+  /* 21 u - 48 + 28 u^-1 */
+  /* 21 u - 48 + 28 u_inv */
+  return 21.f * u - 48.f + 28.f * u_inv;
 }
 
 __attribute__((const)) INLINE static float D_soft_9(const float u,
@@ -233,10 +235,10 @@ __attribute__((const)) INLINE static float D_soft_9(const float u,
   error("Invalid choice of softening kernel shape");
 #endif
 
-  /* -315 u^-1 + 420 u^-3 */
-  /* -315 u_inv + 420 u_inv^3 */
-  float phi = -420.f * u_inv;
-  phi = phi * u_inv + 315.f;
+  /* (1/3) u^-1 - (4/3) u^-3 */
+  /* (1/3) u_inv - (4/3) u_inv^3 */
+  float phi = 1.3333333f * u_inv;
+  phi = phi * u_inv + 0.3333333f;
   phi = phi * u_inv;
 
   return phi;
diff --git a/theory/Multipoles/potential_derivatives.tex b/theory/Multipoles/potential_derivatives.tex
index 14ddbb9792b72e3e815f8362858785f6e889192c..e3e0ead795def8ea3a4f8ae6639b54736440b823 100644
--- a/theory/Multipoles/potential_derivatives.tex
+++ b/theory/Multipoles/potential_derivatives.tex
@@ -98,7 +98,7 @@ truncated an softened gravity field $\varphi (\mathbf{r}, r_s, H)
 \begin{align}
   \mathsf{\tilde{D}}_{5}(r, r_s, H) =
   \left\lbrace\begin{array}{rcl}
-  \left(-105u^3 + 360u^2 - 420u + 168\right)\times  H^{-5}& \mbox{if} & u < 1,\\
+  \left(-35u^3 + 120u^2 - 140u + 56\right)\times  H^{-5}& \mbox{if} & u < 1,\\
   %3\times r^{-5} & \mbox{if} & u \geq 1,
   \left(r^2\chi''(r, r_s) - 3r\chi'(r, r_s) + 3\chi(r, r_s) \right)\times r^{-5} & \mbox{if} & u \geq 1, 
   \end{array}
@@ -108,7 +108,7 @@ truncated an softened gravity field $\varphi (\mathbf{r}, r_s, H)
 \begin{align}
   \mathsf{\tilde{D}}_{7}(r, r_s, H) =
   \left\lbrace\begin{array}{rcl}
-  -\left(315u - 720 + 420u^{-1}\right)\times  H^{-7} & \mbox{if} & u < 1,\\
+  -\left(21u - 48 + 28u^{-1}\right)\times  H^{-7} & \mbox{if} & u < 1,\\
   %-15\times r^{-7} & \mbox{if} & u \geq 1,
   \left(r^3\chi^{(3)}(r, r_s) - 6r^2\chi''(r, r_s)+15r\chi'(r, r_s)-15\chi(r, r_s)\right) \times r^{-7} & \mbox{if} & u \geq 1, 
   \end{array}
@@ -118,7 +118,7 @@ truncated an softened gravity field $\varphi (\mathbf{r}, r_s, H)
 \begin{align}
   \mathsf{\tilde{D}}_{9}(r, r_s, H) =
   \left\lbrace\begin{array}{rcl}
-  \left(-315u^{-1} + 420u^{-3}\right)\times  H^{-9}& \mbox{if} & u < 1,\\
+  \left(-3u^{-1} + 4u^{-3}\right)\times  H^{-9}& \mbox{if} & u < 1,\\
   %105\times r^{-9} & \mbox{if} & u \geq 1.
   \left(r^4\chi^{(4)}(r, r_s) - 10r^3\chi^{(3)} + 45r^2\chi''(r, r_s) - 105r\chi'(r, r_s) + 105\chi(r, r_s) \right) \times r^{-9} & \mbox{if} & u \geq 1
   \end{array}
@@ -128,7 +128,7 @@ truncated an softened gravity field $\varphi (\mathbf{r}, r_s, H)
 \begin{align}
   \mathsf{\tilde{D}}_{11}(r, r_s, H) =
   \left\lbrace\begin{array}{rcl}
-  -\left(315u^{-3} - 1260u^{-5}\right)\times  H^{-11}& \mbox{if} & u < 1,\\
+  -\left(\frac{1}{3}u^{-3} - \frac{4}{3}u^{-5}\right)\times  H^{-11}& \mbox{if} & u < 1,\\
   %-945\times r^{-11} & \mbox{if} & u \geq 1.
   \left(r^5\chi^{(5)}(r, r_s) - 15r^4\chi^{(4)}(r, r_s) + 105r^3\chi^{(3)}(r, r_s) - 420r^2\chi''(r, r_s) + 945r \chi'(r, r_s) - 945\chi(r, r_s)\right) \times r^{-11} & \mbox{if} & u \geq 1.
   \end{array}
@@ -262,7 +262,7 @@ r_y \mathsf{\tilde{D}}_{5}(r, r_s, H) \nonumber
   r_x^2 r_y^2 r_z \mathsf{\tilde{D}}_{11}(r, r_s, H) +
   r_x^2 r_z \mathsf{\tilde{D}}_{9}(r, r_s, H) +
   r_y^2 r_z \mathsf{\tilde{D}}_{9}(r, r_s, H) +
-  r_z \mathsf{\tilde{D}}_{y}(r, r_s, H)
+  r_z \mathsf{\tilde{D}}_{7}(r, r_s, H)
   \nonumber
 \end{align}