diff --git a/doc/math.lyx b/doc/math.lyx
index f14ebc66f..6d7a5e318 100644
--- a/doc/math.lyx
+++ b/doc/math.lyx
@@ -5090,29 +5090,22 @@ Derivative of Adjoint
 
 \begin_layout Standard
 Consider 
-\begin_inset Formula $f:SE(3)\rightarrow\mathbb{R}^{6}$
+\begin_inset Formula $f:SE(3)\times\mathbb{R}^{6}\rightarrow\mathbb{R}^{6}$
 \end_inset
 
  is defined as 
-\begin_inset Formula $f(T)=Ad_{T}y$
-\end_inset
-
- for some constant 
-\begin_inset Formula $y=\begin{bmatrix}\omega_{y}\\
-v_{y}
-\end{bmatrix}$
+\begin_inset Formula $f(T,\xi_{b})=Ad_{T}\hat{\xi}_{b}$
 \end_inset
 
 .
- Defining 
-\begin_inset Formula $\xi=\begin{bmatrix}\omega\\
-v
-\end{bmatrix}$
-\end_inset
+ The derivative is notated (see 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Derivatives-of-Actions"
+plural "false"
+caps "false"
+noprefix "false"
 
- for the derivative notation (w.r.t.
- pose 
-\begin_inset Formula $T$
 \end_inset
 
 ):
@@ -5121,68 +5114,17 @@ v
 \begin_layout Standard
 \begin_inset Formula 
 \[
-f'(T)=\frac{\partial Ad_{T}y}{\partial\xi}=\begin{bmatrix}\frac{\partial f}{\omega} & \frac{\partial f}{v}\end{bmatrix}
+Df_{(T,y)}(\xi,\delta y)=D_{1}f_{(T,y)}(\xi)+D_{2}f_{(T,y)}(\delta y)
 \]
 
 \end_inset
 
-Then we can compute 
-\begin_inset Formula $f'(T)$
+First, computing 
+\begin_inset Formula $D_{2}f_{(T,\xi_{b})}(\xi_{b})$
 \end_inset
 
- by considering the rotation and translation separately.
- To reduce confusion with 
-\begin_inset Formula $\omega_{y},v_{y}$
-\end_inset
-
-, we will use 
-\begin_inset Formula $R,t$
-\end_inset
-
- to denote the rotation and translation of 
-\begin_inset Formula $T\exp\xi$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\[
-\frac{\partial}{\partial\omega}\begin{bmatrix}R & 0\\{}
-[t]_{\times}R & R
-\end{bmatrix}\begin{bmatrix}\omega_{y}\\
-v_{y}
-\end{bmatrix}=\frac{\partial}{\partial\omega}\begin{bmatrix}R\omega_{y}\\{}
-[t]_{\times}R\omega_{y}+Rv_{y}
-\end{bmatrix}=\begin{bmatrix}-R[\omega_{y}]_{\times}\\
--[t]_{\times}R[w_{y}]_{\times}-R[v_{y}]_{\times}
-\end{bmatrix}
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\[
-\frac{\partial}{\partial t}\begin{bmatrix}R & 0\\{}
-[t]_{\times}R & R
-\end{bmatrix}\begin{bmatrix}\omega\\
-v
-\end{bmatrix}=\frac{\partial}{\partial t}\begin{bmatrix}R\omega_{y}\\{}
-[t]_{\times}R\omega_{y}+Rv_{y}
-\end{bmatrix}=\begin{bmatrix}0\\
--[R\omega_{y}]
-\end{bmatrix}
-\]
-
-\end_inset
-
-Applying chain rule for the translation, 
-\begin_inset Formula $\frac{\partial}{\partial v}=\frac{\partial}{\partial t}\frac{\partial t}{\partial v}$
+ is easy, as its matrix is simply 
+\begin_inset Formula $Ad_{T}$
 \end_inset
 
 :
@@ -5191,76 +5133,139 @@ Applying chain rule for the translation,
 \begin_layout Standard
 \begin_inset Formula 
 \[
-\frac{\partial}{\partial v}\begin{bmatrix}R & 0\\{}
-[t]_{\times}R & R
-\end{bmatrix}\begin{bmatrix}\omega_{y}\\
-v_{y}
-\end{bmatrix}=\begin{bmatrix}0\\
--[R\omega_{y}]_{\times}
-\end{bmatrix}R=\begin{bmatrix}0\\
--[R\omega_{y}]_{\times}R
-\end{bmatrix}=\begin{bmatrix}0\\
--R[\omega_{y}]_{\times}
-\end{bmatrix}
+f(T,\xi_{b}+\delta\xi_{b})=Ad_{T}(\widehat{\xi_{b}+\delta\xi_{b}})=Ad_{T}(\hat{\xi}_{b})+Ad_{T}(\delta\hat{\xi}_{b})
 \]
 
 \end_inset
 
-The simplification 
-\family roman
-\series medium
-\shape up
-\size normal
-\emph off
-\bar no
-\strikeout off
-\xout off
-\uuline off
-\uwave off
-\noun off
-\color none
 
-\begin_inset Formula $[R\omega_{y}]_{\times}R=R[\omega_{y}]_{\times}$
-\end_inset
-
- can be derived by considering the result for when 
-\begin_inset Formula $\omega_{y}$
-\end_inset
-
- is each of the 3 standard basis vectors 
-\begin_inset Formula $\hat{e}_{i}$
-\end_inset
-
-: 
-\begin_inset Formula $-[R\hat{e}_{i}]_{\times}R$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Now putting together the rotation and translation:
 \end_layout
 
 \begin_layout Standard
 \begin_inset Formula 
 \[
-f'(T)=\frac{\partial Ad_{T}y}{\partial\xi}=\begin{bmatrix}\frac{\partial f}{\omega} & \frac{\partial f}{v}\end{bmatrix}=\begin{bmatrix}-R[\omega_{y}]_{\times} & 0\\
--[t]_{\times}R[w_{y}]_{\times}-R[v_{y}]_{\times} & -R[\omega_{y}]_{\times}
-\end{bmatrix}
+D_{2}f_{(T,\xi_{b})}(\xi_{b})=Ad_{T}
 \]
 
 \end_inset
 
+To compute 
+\begin_inset Formula $D_{1}f_{(T,\xi_{b})}(\xi_{b})$
+\end_inset
 
-\end_layout
-
-\begin_layout Standard
-We can apply a similar procedure to compute the derivative of 
-\begin_inset Formula $Ad_{T}^{T}y$
+, we'll first define 
+\begin_inset Formula $g(T,\xi)\triangleq T\exp\hat{\xi}$
 \end_inset
 
 .
+ From Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Derivatives-of-Actions"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{align*}
+D_{2}g_{(T,\xi)}(\xi) & =T\hat{\xi}\\
+D_{2}g_{(T,\xi)}^{-1}(\xi) & =-\hat{\xi}T^{-1}
+\end{align*}
+
+\end_inset
+
+Now we can use the definition of the Adjoint representation 
+\begin_inset Formula $Ad_{g}\hat{\xi}=g\hat{\xi}g^{-1}$
+\end_inset
+
+ (aka conjugation by 
+\begin_inset Formula $g$
+\end_inset
+
+) then apply product rule and simplify:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{align*}
+D_{1}f_{(T,\xi_{b})}(\xi)=D_{1}\left(Ad_{T\exp(\hat{\xi})}\hat{\xi}_{b}\right)(\xi) & =D_{1}\left(g\hat{\xi}g^{-1}\right)(\xi)\\
+ & =\left(D_{2}g_{(T,\xi)}(\xi)\right)\hat{\xi}g^{-1}(T,0)+g(T,0)\hat{\xi}\left(D_{2}g_{(T,\xi)}^{-1}(\xi)\right)\\
+ & =T\hat{\xi}\hat{\xi}_{b}T^{-1}-T\hat{\xi}_{b}\hat{\xi}T^{-1}\\
+ & =T\left(\hat{\xi}\hat{\xi}_{b}-\hat{\xi}_{b}\hat{\xi}\right)T^{-1}\\
+ & =-Ad_{T}(ad_{\xi_{b}}\hat{\xi})\\
+D_{1}F_{(T,\xi_{b})} & =-(Ad_{T})(ad_{\hat{\xi}_{b}})
+\end{align*}
+
+\end_inset
+
+An alternative, perhaps more intuitive way of deriving this would be to
+ use the fact that the derivative at the origin 
+\begin_inset Formula $D_{1}Ad_{I}\hat{\xi}_{b}=ad_{\hat{\xi}_{b}}$
+\end_inset
+
+ by definition of the adjoint 
+\begin_inset Formula $ad_{\xi}$
+\end_inset
+
+.
+ Then applying the property 
+\begin_inset Formula $Ad_{AB}=Ad_{A}Ad_{B}$
+\end_inset
+
+,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+D_{1}Ad_{T}\hat{\xi}_{b}(\xi)=D_{1}Ad_{T*I}\hat{\xi}_{b}(\xi)=Ad_{T}\left(D_{1}Ad_{I}\hat{\xi}_{b}(\xi)\right)=Ad_{T}\left(ad_{\hat{\xi}_{b}}(\xi)\right)
+\]
+
+\end_inset
+
+It's difficult to apply a similar procedure to compute the derivative of
+ 
+\begin_inset Formula $Ad_{T}^{T}\hat{\xi}_{b}^{*}$
+\end_inset
+
+ (note the
+\begin_inset Formula $^{*}$
+\end_inset
+
+ denoting that we are now in the dual space) because 
+\begin_inset Formula $Ad_{T}^{T}$
+\end_inset
+
+ cannot be naturally defined as a conjugation so we resort to crunching
+ through the algebra.
+ The details are omitted but the result is a form vaguely resembling (but
+ not quite) the 
+\begin_inset Formula $ad(Ad_{T}^{T}\hat{\xi}_{b}^{*})$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{align*}
+\begin{bmatrix}\omega_{T}\\
+v_{T}
+\end{bmatrix}^{*} & \triangleq Ad_{T}^{T}\hat{\xi}_{b}^{*}\\
+D_{1}Ad_{T}^{T}\hat{\xi}_{b}^{*}(\xi) & =\begin{bmatrix}\hat{\omega}_{T} & \hat{v}_{T}\\
+\hat{v}_{T} & 0
+\end{bmatrix}
+\end{align*}
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Subsection
diff --git a/doc/math.pdf b/doc/math.pdf
index 06dec078d..71f9dadc6 100644
Binary files a/doc/math.pdf and b/doc/math.pdf differ
diff --git a/gtsam/geometry/Pose3.cpp b/gtsam/geometry/Pose3.cpp
index ed605d8b6..4bfb574b1 100644
--- a/gtsam/geometry/Pose3.cpp
+++ b/gtsam/geometry/Pose3.cpp
@@ -67,88 +67,41 @@ Matrix6 Pose3::AdjointMap() const {
 // Calculate AdjointMap applied to xi_b, with Jacobians
 Vector6 Pose3::Adjoint(const Vector6& xi_b, OptionalJacobian<6, 6> H_pose,
                        OptionalJacobian<6, 6> H_xib) const {
-  //  Ad   *   xi   =   [   R   0    .  [w
-  //                      [t]R  R ]      v]
-  // Declarations, aliases, and intermediate Jacobians easy to compute now
-  Vector6 result; // = AdjointMap() * xi
-  auto Rw = result.head<3>();
-  const auto &w = xi_b.head<3>(), &v = xi_b.tail<3>();
-  Matrix3 Rw_H_R, Rv_H_R, pRw_H_Rw;
-  const Matrix3 R = R_.matrix();
-  const Matrix3 &Rw_H_w = R;
-  const Matrix3 &Rv_H_v = R;
-
-  // Calculations
-  Rw = R_.rotate(w, H_pose ? &Rw_H_R : nullptr /*, Rw_H_w */);
-  const Vector3 Rv = R_.rotate(v, H_pose ? &Rv_H_R : nullptr /*, Rv_H_v */);
-  const Vector3 pRw = cross(t_, Rw, boost::none /* pRw_H_t */, pRw_H_Rw);
-  result.tail<3>() = pRw + Rv;
+  const Matrix6 Ad = AdjointMap();
 
   // Jacobians
-  if (H_pose) {
-    // pRw_H_posev = pRw_H_t * R = [Rw]x * R = R * [w]x = Rw_H_R
-    //    where [ ]x denotes the skew-symmetric operator.
-    // See docs/math.pdf for more details.
-    const Matrix3 &pRw_H_posev = Rw_H_R;
-    *H_pose = (Matrix6() << Rw_H_R, /*               */ Z_3x3,  //
-               /*        */ pRw_H_Rw * Rw_H_R + Rv_H_R, pRw_H_posev)
-                  .finished();
-  }
-  if (H_xib) {
-    // This is just equal to AdjointMap() but we can reuse pRw_H_Rw = [t]x
-    *H_xib = (Matrix6() << Rw_H_w, /*      */ Z_3x3,
-              /*        */ pRw_H_Rw * Rw_H_w, Rv_H_v)
-                 .finished();
-  }
+  // D1 Ad_T(xi_b) = D1 Ad_T Ad_I(xi_b) = Ad_T * D1 Ad_I(xi_b) = Ad_T * ad_xi_b
+  // D2 Ad_T(xi_b) = Ad_T
+  // See docs/math.pdf for more details.
+  // In D1 calculation, we could be more efficient by writing it out, but do not
+  // for readability
+  if (H_pose) *H_pose = -Ad * adjointMap(xi_b);
+  if (H_xib) *H_xib = Ad;
 
-  // Return - we computed result manually but it should be = AdjointMap() * xi
-  return result;
+  return Ad * xi_b;
 }
 
 /* ************************************************************************* */
 /// The dual version of Adjoint
 Vector6 Pose3::AdjointTranspose(const Vector6& x, OptionalJacobian<6, 6> H_pose,
                                 OptionalJacobian<6, 6> H_x) const {
-  //  Ad^T   *   xi   =   [ R^T  R^T.[-t]  .  [w
-  //                         0     R^T ]       v]
-  // Declarations, aliases, and intermediate Jacobians easy to compute now
-  Vector6 result; // = AdjointMap().transpose() * x
-  const Vector3 &w = x.head<3>(), &v = x.tail<3>();
-  auto Rv = result.tail<3>();
-  Matrix3 Rw_H_R, Rv_H_R, Rtv_H_R;
-  const Matrix3 Rtranspose = R_.matrix().transpose();
-  const Matrix3 &Rw_H_w = Rtranspose;
-  const Matrix3 &Rv_H_v = Rtranspose;
-  const Matrix3 &Rtv_H_tv = Rtranspose;
-  const Matrix3 tv_H_v = skewSymmetric(t_);
-
-  // Calculations
-  const Vector3 Rw = R_.unrotate(w, H_pose ? &Rw_H_R : nullptr /*, Rw_H_w */);
-  Rv = R_.unrotate(v, H_pose ? &Rv_H_R : nullptr /*, Rv_H_v */);
-  const Vector3 tv = cross(t_, v, boost::none /* tv_H_t */, tv_H_v);
-  const Vector3 Rtv =
-      R_.unrotate(tv, H_pose ? &Rtv_H_R : nullptr /*, Rtv_H_tv */);
-  result.head<3>() = Rw - Rtv;
+  const Matrix6 &AdT = AdjointMap().transpose();
+  const Vector6 &AdTx = AdT * x;
 
   // Jacobians
+  // See docs/math.pdf for more details.
   if (H_pose) {
-    // Rtv_H_posev = -Rtv_H_tv * tv_H_t * R = -R' * -[v]x * R = -[R'v]x = Rv_H_R
-    //    where [ ]x denotes the skew-symmetric operator.
-    // See docs/math.pdf for more details.
-    const Matrix3 &Rtv_H_posev = Rv_H_R;
-    *H_pose = (Matrix6() << Rw_H_R - Rtv_H_R, Rtv_H_posev,
-               /*        */ Rv_H_R, /*     */ Z_3x3)
+    const auto &w_T_hat = skewSymmetric(AdTx.head<3>()),
+               &v_T_hat = skewSymmetric(AdTx.tail<3>());
+    *H_pose = (Matrix6() << w_T_hat, v_T_hat,  //
+               /*        */ v_T_hat, Z_3x3)
                   .finished();
   }
   if (H_x) {
-    // This is just equal to AdjointMap().transpose() but we can reuse [t]x
-    *H_x = (Matrix6() << Rw_H_w, -Rtv_H_tv * tv_H_v,
-            /*        */ Z_3x3, Rv_H_v)
-               .finished();
+    *H_x = AdT;
   }
 
-  // Return - this should be equivalent to AdjointMap().transpose() * xi
-  return result;
+  return AdTx;
 }
 
 /* ************************************************************************* */