diff --git a/.travis.yml b/.travis.yml
--- a/.travis.yml
+++ b/.travis.yml
@@ -8,7 +8,7 @@
 
 install:
   - cabal install packunused packdeps
-  - cabal configure -flib-Werror $mode
+  - cabal configure $mode
   - cabal build --ghc-options=-ddump-minimal-imports
 
 script:
diff --git a/CHANGELOG.markdown b/CHANGELOG.markdown
--- a/CHANGELOG.markdown
+++ b/CHANGELOG.markdown
@@ -1,3 +1,8 @@
+4.2.2
+-----
+* *Major* bug fix for `grads`, `jacobians`, and anything that uses `Sparse` mode in `Numeric.AD`. Derivatives after the first two
+  were previously incorrect.
+
 4.2.1.1
 -------
 * Support `nats` version 1
diff --git a/LICENSE b/LICENSE
--- a/LICENSE
+++ b/LICENSE
@@ -1,4 +1,4 @@
-Copyright (c) 2010, Edward Kmett
+Copyright (c) 2010-2015, Edward Kmett
           (c) 2008-2009 Barak A. Pearlmutter and Jeffrey Mark Siskind
 
 All rights reserved.
diff --git a/ad.cabal b/ad.cabal
--- a/ad.cabal
+++ b/ad.cabal
@@ -1,8 +1,8 @@
 name:         ad
-version:      4.2.1.1
+version:      4.2.2
 license:      BSD3
 license-File: LICENSE
-copyright:    (c) Edward Kmett 2010-2014,
+copyright:    (c) Edward Kmett 2010-2015,
               (c) Barak Pearlmutter and Jeffrey Mark Siskind 2008-2009
 author:       Edward Kmett
 maintainer:   ekmett@gmail.com
@@ -71,10 +71,6 @@
     .
     * @0@ means that the resulting derivative list is padded with 0s at the end.
 
-flag lib-Werror
-  default: False
-  manual: True
-
 source-repository head
   type: git
   location: git://github.com/ekmett/ad.git
@@ -151,10 +147,7 @@
   other-modules:
     Numeric.AD.Internal.Combinators
 
-  if flag(lib-Werror)
-    ghc-options: -Werror
-  else
-    ghc-options: -Wall
+  ghc-options: -Wall
 
   ghc-options: -fspec-constr -fdicts-cheap -O2
 
diff --git a/src/Numeric/AD.hs b/src/Numeric/AD.hs
--- a/src/Numeric/AD.hs
+++ b/src/Numeric/AD.hs
@@ -4,7 +4,7 @@
 {-# LANGUAGE PatternGuards #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Halley.hs b/src/Numeric/AD/Halley.hs
--- a/src/Numeric/AD/Halley.hs
+++ b/src/Numeric/AD/Halley.hs
@@ -2,7 +2,7 @@
 {-# LANGUAGE ScopedTypeVariables #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Internal/Combinators.hs b/src/Numeric/AD/Internal/Combinators.hs
--- a/src/Numeric/AD/Internal/Combinators.hs
+++ b/src/Numeric/AD/Internal/Combinators.hs
@@ -2,7 +2,7 @@
 {-# OPTIONS_HADDOCK not-home #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Internal/Dense.hs b/src/Numeric/AD/Internal/Dense.hs
--- a/src/Numeric/AD/Internal/Dense.hs
+++ b/src/Numeric/AD/Internal/Dense.hs
@@ -11,7 +11,7 @@
 
 -----------------------------------------------------------------------------
 -- |
--- Copyright   : (c) Edward Kmett 2010-2014
+-- Copyright   : (c) Edward Kmett 2010-2015
 -- License     : BSD3
 -- Maintainer  : ekmett@gmail.com
 -- Stability   : experimental
diff --git a/src/Numeric/AD/Internal/Forward.hs b/src/Numeric/AD/Internal/Forward.hs
--- a/src/Numeric/AD/Internal/Forward.hs
+++ b/src/Numeric/AD/Internal/Forward.hs
@@ -11,7 +11,7 @@
 
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Internal/Forward/Double.hs b/src/Numeric/AD/Internal/Forward/Double.hs
--- a/src/Numeric/AD/Internal/Forward/Double.hs
+++ b/src/Numeric/AD/Internal/Forward/Double.hs
@@ -10,7 +10,7 @@
 
 -----------------------------------------------------------------------------
 ---- |
----- Copyright   :  (c) Edward Kmett 2010-2014
+---- Copyright   :  (c) Edward Kmett 2010-2015
 ---- License     :  BSD3
 ---- Maintainer  :  ekmett@gmail.com
 ---- Stability   :  experimental
diff --git a/src/Numeric/AD/Internal/Identity.hs b/src/Numeric/AD/Internal/Identity.hs
--- a/src/Numeric/AD/Internal/Identity.hs
+++ b/src/Numeric/AD/Internal/Identity.hs
@@ -8,7 +8,7 @@
 
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Internal/Kahn.hs b/src/Numeric/AD/Internal/Kahn.hs
--- a/src/Numeric/AD/Internal/Kahn.hs
+++ b/src/Numeric/AD/Internal/Kahn.hs
@@ -12,7 +12,7 @@
 
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Internal/On.hs b/src/Numeric/AD/Internal/On.hs
--- a/src/Numeric/AD/Internal/On.hs
+++ b/src/Numeric/AD/Internal/On.hs
@@ -10,7 +10,7 @@
 
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Internal/Or.hs b/src/Numeric/AD/Internal/Or.hs
--- a/src/Numeric/AD/Internal/Or.hs
+++ b/src/Numeric/AD/Internal/Or.hs
@@ -10,7 +10,7 @@
 
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2014
+-- Copyright   :  (c) Edward Kmett 2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Internal/Reverse.hs b/src/Numeric/AD/Internal/Reverse.hs
--- a/src/Numeric/AD/Internal/Reverse.hs
+++ b/src/Numeric/AD/Internal/Reverse.hs
@@ -14,7 +14,7 @@
 
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2012-2014
+-- Copyright   :  (c) Edward Kmett 2012-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
@@ -24,7 +24,7 @@
 --
 -- This version uses @Data.Reflection@ to find and update the tape.
 --
--- This is asymptotically faster than using @Reverse@, which
+-- This is asymptotically faster than using @Kahn@, which
 -- is forced to reify and topologically sort the graph, but it requires
 -- a fairly expensive rendezvous during construction when updated using
 -- multiple threads.
diff --git a/src/Numeric/AD/Internal/Sparse.hs b/src/Numeric/AD/Internal/Sparse.hs
--- a/src/Numeric/AD/Internal/Sparse.hs
+++ b/src/Numeric/AD/Internal/Sparse.hs
@@ -10,7 +10,18 @@
 {-# LANGUAGE UndecidableInstances #-}
 {-# OPTIONS_GHC -fno-warn-name-shadowing #-}
 {-# OPTIONS_HADDOCK not-home #-}
-
+-----------------------------------------------------------------------------
+-- |
+-- Copyright   :  (c) Edward Kmett 2010-2015
+-- License     :  BSD3
+-- Maintainer  :  ekmett@gmail.com
+-- Stability   :  experimental
+-- Portability :  GHC only
+--
+-- Unsafe and often partial combinators intended for internal usage.
+--
+-- Handle with care.
+-----------------------------------------------------------------------------
 module Numeric.AD.Internal.Sparse
   ( Index(..)
   , emptyIndex
@@ -75,10 +86,10 @@
 times :: Num a => Sparse a -> Int -> Sparse a -> Sparse a
 times Zero _ _ = Zero
 times _ _ Zero = Zero
-times (Sparse a as) n (Sparse b bs) = Sparse (a * b) $
+times a@(Sparse pa da) n b@(Sparse pb db) = Sparse (pa * pb) $
   unionWith (+)
-    (fmap (^* b) (dropMap n as))
-    (fmap (a *^) (dropMap n bs))
+    (fmap (* b) (dropMap n da))
+    (fmap (a *) (dropMap n db))
 {-# INLINE times #-}
 
 vars :: (Traversable f, Num a) => f a -> f (Sparse a)
@@ -112,32 +123,6 @@
   go ix i = partial (indices ix') as :< (go ix' <$> fns) where
     ix' = addToIndex i ix
 {-# INLINE ds #-}
-
-{-
-vvars :: Num a => Vector a -> Vector (AD Sparse a)
-vvars = Vector.imap (\n a -> AD $ Sparse a $ singleton n $ auto 1)
-{-# INLINE vvars #-}
-
-vapply :: Num a => (Vector (AD Sparse a) -> b) -> Vector a -> b
-vapply f = f . vvars
-{-# INLINE vapply #-}
-
-
-vd :: Num a => Int -> AD Sparse a -> Vector a
-vd n (AD (Sparse _ da)) = Vector.generate n $ \i -> maybe 0 primal $ lookup i da
-{-# INLINE vd #-}
-
-vd' :: Num a => Int -> AD Sparse a -> (a, Vector a)
-vd' n (AD (Sparse a da)) = (a , Vector.generate n $ \i -> maybe 0 primal $ lookup i da)
-{-# INLINE vd' #-}
-
-vds :: Num a => Int -> AD Sparse a -> Cofree Vector a
-vds n (AD as@(Sparse a _)) = a :< Vector.generate n (go emptyIndex)
-    where
-        go ix i = partial (indices ix') as :< Vector.generate n (go ix')
-            where ix' = addToIndex i ix
-{-# INLINE vds #-}
--}
 
 partial :: Num a => [Int] -> Sparse a -> a
 partial []     (Sparse a _)  = a
diff --git a/src/Numeric/AD/Internal/Tower.hs b/src/Numeric/AD/Internal/Tower.hs
--- a/src/Numeric/AD/Internal/Tower.hs
+++ b/src/Numeric/AD/Internal/Tower.hs
@@ -11,7 +11,7 @@
 
 -----------------------------------------------------------------------------
 -- |
--- Copyright   : (c) Edward Kmett 2010-2014
+-- Copyright   : (c) Edward Kmett 2010-2015
 -- License     : BSD3
 -- Maintainer  : ekmett@gmail.com
 -- Stability   : experimental
diff --git a/src/Numeric/AD/Internal/Type.hs b/src/Numeric/AD/Internal/Type.hs
--- a/src/Numeric/AD/Internal/Type.hs
+++ b/src/Numeric/AD/Internal/Type.hs
@@ -3,7 +3,7 @@
 {-# LANGUAGE GeneralizedNewtypeDeriving #-}
 -----------------------------------------------------------------------------
 ---- |
----- Copyright   :  (c) Edward Kmett 2010-2014
+---- Copyright   :  (c) Edward Kmett 2010-2015
 ---- License     :  BSD3
 ---- Maintainer  :  ekmett@gmail.com
 ---- Stability   :  experimental
diff --git a/src/Numeric/AD/Jacobian.hs b/src/Numeric/AD/Jacobian.hs
--- a/src/Numeric/AD/Jacobian.hs
+++ b/src/Numeric/AD/Jacobian.hs
@@ -11,7 +11,7 @@
 {-# LANGUAGE UndecidableInstances #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Jet.hs b/src/Numeric/AD/Jet.hs
--- a/src/Numeric/AD/Jet.hs
+++ b/src/Numeric/AD/Jet.hs
@@ -8,7 +8,7 @@
 #endif
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Mode.hs b/src/Numeric/AD/Mode.hs
--- a/src/Numeric/AD/Mode.hs
+++ b/src/Numeric/AD/Mode.hs
@@ -10,7 +10,7 @@
 {-# LANGUAGE UndecidableInstances #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Mode/Forward.hs b/src/Numeric/AD/Mode/Forward.hs
--- a/src/Numeric/AD/Mode/Forward.hs
+++ b/src/Numeric/AD/Mode/Forward.hs
@@ -2,7 +2,7 @@
 {-# LANGUAGE Rank2Types #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Mode/Forward/Double.hs b/src/Numeric/AD/Mode/Forward/Double.hs
--- a/src/Numeric/AD/Mode/Forward/Double.hs
+++ b/src/Numeric/AD/Mode/Forward/Double.hs
@@ -1,4 +1,17 @@
 {-# LANGUAGE Rank2Types #-}
+-----------------------------------------------------------------------------
+-- |
+-- Copyright   :  (c) Edward Kmett 2010-2015
+-- License     :  BSD3
+-- Maintainer  :  ekmett@gmail.com
+-- Stability   :  experimental
+-- Portability :  GHC only
+--
+-- Forward Mode AD specialized to `Double`. This enables the entire structure
+-- to be unboxed.
+--
+-----------------------------------------------------------------------------
+
 module Numeric.AD.Mode.Forward.Double
   ( AD
   , ForwardDouble
diff --git a/src/Numeric/AD/Mode/Kahn.hs b/src/Numeric/AD/Mode/Kahn.hs
--- a/src/Numeric/AD/Mode/Kahn.hs
+++ b/src/Numeric/AD/Mode/Kahn.hs
@@ -7,7 +7,7 @@
 {-# LANGUAGE UndecidableInstances #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Mode/Reverse.hs b/src/Numeric/AD/Mode/Reverse.hs
--- a/src/Numeric/AD/Mode/Reverse.hs
+++ b/src/Numeric/AD/Mode/Reverse.hs
@@ -8,7 +8,7 @@
 {-# LANGUAGE ScopedTypeVariables #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Mode/Sparse.hs b/src/Numeric/AD/Mode/Sparse.hs
--- a/src/Numeric/AD/Mode/Sparse.hs
+++ b/src/Numeric/AD/Mode/Sparse.hs
@@ -1,7 +1,7 @@
 {-# LANGUAGE Rank2Types #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   : (c) Edward Kmett 2010-2014
+-- Copyright   : (c) Edward Kmett 2010-2015
 -- License     : BSD3
 -- Maintainer  : ekmett@gmail.com
 -- Stability   : experimental
diff --git a/src/Numeric/AD/Mode/Tower.hs b/src/Numeric/AD/Mode/Tower.hs
--- a/src/Numeric/AD/Mode/Tower.hs
+++ b/src/Numeric/AD/Mode/Tower.hs
@@ -1,7 +1,7 @@
 {-# LANGUAGE Rank2Types #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   : (c) Edward Kmett 2010-2014
+-- Copyright   : (c) Edward Kmett 2010-2015
 -- License     : BSD3
 -- Maintainer  : ekmett@gmail.com
 -- Stability   : experimental
diff --git a/src/Numeric/AD/Newton.hs b/src/Numeric/AD/Newton.hs
--- a/src/Numeric/AD/Newton.hs
+++ b/src/Numeric/AD/Newton.hs
@@ -5,7 +5,7 @@
 {-# LANGUAGE TypeFamilies #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Rank1/Forward.hs b/src/Numeric/AD/Rank1/Forward.hs
--- a/src/Numeric/AD/Rank1/Forward.hs
+++ b/src/Numeric/AD/Rank1/Forward.hs
@@ -1,7 +1,7 @@
 {-# LANGUAGE CPP #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Rank1/Halley.hs b/src/Numeric/AD/Rank1/Halley.hs
--- a/src/Numeric/AD/Rank1/Halley.hs
+++ b/src/Numeric/AD/Rank1/Halley.hs
@@ -1,6 +1,6 @@
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Rank1/Kahn.hs b/src/Numeric/AD/Rank1/Kahn.hs
--- a/src/Numeric/AD/Rank1/Kahn.hs
+++ b/src/Numeric/AD/Rank1/Kahn.hs
@@ -7,7 +7,7 @@
 {-# LANGUAGE UndecidableInstances #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Rank1/Newton.hs b/src/Numeric/AD/Rank1/Newton.hs
--- a/src/Numeric/AD/Rank1/Newton.hs
+++ b/src/Numeric/AD/Rank1/Newton.hs
@@ -4,7 +4,7 @@
 {-# LANGUAGE TypeFamilies #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   :  (c) Edward Kmett 2010-2014
+-- Copyright   :  (c) Edward Kmett 2010-2015
 -- License     :  BSD3
 -- Maintainer  :  ekmett@gmail.com
 -- Stability   :  experimental
diff --git a/src/Numeric/AD/Rank1/Sparse.hs b/src/Numeric/AD/Rank1/Sparse.hs
--- a/src/Numeric/AD/Rank1/Sparse.hs
+++ b/src/Numeric/AD/Rank1/Sparse.hs
@@ -1,6 +1,6 @@
 -----------------------------------------------------------------------------
 -- |
--- Copyright   : (c) Edward Kmett 2010-2014
+-- Copyright   : (c) Edward Kmett 2010-2015
 -- License     : BSD3
 -- Maintainer  : ekmett@gmail.com
 -- Stability   : experimental
diff --git a/src/Numeric/AD/Rank1/Tower.hs b/src/Numeric/AD/Rank1/Tower.hs
--- a/src/Numeric/AD/Rank1/Tower.hs
+++ b/src/Numeric/AD/Rank1/Tower.hs
@@ -2,7 +2,7 @@
 {-# LANGUAGE BangPatterns #-}
 -----------------------------------------------------------------------------
 -- |
--- Copyright   : (c) Edward Kmett 2010-2014
+-- Copyright   : (c) Edward Kmett 2010-2015
 -- License     : BSD3
 -- Maintainer  : ekmett@gmail.com
 -- Stability   : experimental
@@ -29,89 +29,107 @@
   , diffsF  -- answer and all derivatives of (a -> f a)
   , diffs0F -- zero padded derivatives of (a -> f a)
   -- * Directional Derivatives
-  , du      -- directional derivative of (a -> a)
-  , du'     -- answer and directional derivative of (a -> a)
-  , dus     -- answer and all directional derivatives of (a -> a)
-  , dus0    -- answer and all zero padded directional derivatives of (a -> a)
-  , duF     -- directional derivative of (a -> f a)
-  , duF'    -- answer and directional derivative of (a -> f a)
-  , dusF    -- answer and all directional derivatives of (a -> f a)
-  , dus0F   -- answer and all zero padded directional derivatives of (a -> a)
+  , du      -- directional derivative of (f a -> a)
+  , du'     -- answer and directional derivative of (f a -> a)
+  , dus     -- answer and all directional derivatives of (f a -> a)
+  , dus0    -- answer and all zero padded directional derivatives of (f a -> a)
+  , duF     -- directional derivative of (f a -> g a)
+  , duF'    -- answer and directional derivative of (f a -> g a)
+  , dusF    -- answer and all directional derivatives of (f a -> g a)
+  , dus0F   -- answer and all zero padded directional derivatives of (f a -> g a)
   ) where
 
 import Control.Applicative ((<$>))
 import Numeric.AD.Internal.Tower
 import Numeric.AD.Mode
 
+-- | Compute the answer and all derivatives of a function @(a -> a)@
 diffs :: Num a => (Tower a -> Tower a) -> a -> [a]
 diffs f a = getADTower $ apply f a
 {-# INLINE diffs #-}
 
+-- | Compute the zero-padded derivatives of a function @(a -> a)@
 diffs0 :: Num a => (Tower a -> Tower a) -> a -> [a]
 diffs0 f a = zeroPad (diffs f a)
 {-# INLINE diffs0 #-}
 
+-- | Compute the answer and all derivatives of a function @(a -> f a)@
 diffsF :: (Functor f, Num a) => (Tower a -> f (Tower a)) -> a -> f [a]
 diffsF f a = getADTower <$> apply f a
 {-# INLINE diffsF #-}
 
+-- | Compute the zero-padded derivatives of a function @(a -> f a)@
 diffs0F :: (Functor f, Num a) => (Tower a -> f (Tower a)) -> a -> f [a]
 diffs0F f a = (zeroPad . getADTower) <$> apply f a
 {-# INLINE diffs0F #-}
 
+-- | @taylor f x@ compute the Taylor series of @f@ around @x@.
 taylor :: Fractional a => (Tower a -> Tower a) -> a -> a -> [a]
 taylor f x dx = go 1 1 (diffs f x) where
   go !n !acc (a:as) = a * acc : go (n + 1) (acc * dx / n) as
   go _ _ [] = []
 
+-- | @taylor0 f x@ compute the Taylor series of @f@ around @x@, zero-padded.
 taylor0 :: Fractional a => (Tower a -> Tower a) -> a -> a -> [a]
 taylor0 f x dx = zeroPad (taylor f x dx)
 {-# INLINE taylor0 #-}
 
+-- | @maclaurin f@ compute the Maclaurin series of @f@
 maclaurin :: Fractional a => (Tower a -> Tower a) -> a -> [a]
 maclaurin f = taylor f 0
 {-# INLINE maclaurin #-}
 
+-- | @maclaurin f@ compute the Maclaurin series of @f@, zero-padded
 maclaurin0 :: Fractional a => (Tower a -> Tower a) -> a -> [a]
 maclaurin0 f = taylor0 f 0
 {-# INLINE maclaurin0 #-}
 
+-- | Compute the first derivative of a function @(a -> a)@
 diff :: Num a => (Tower a -> Tower a) -> a -> a
 diff f = d . diffs f
 {-# INLINE diff #-}
 
+-- | Compute the answer and first derivative of a function @(a -> a)@
 diff' :: Num a => (Tower a -> Tower a) -> a -> (a, a)
 diff' f = d' . diffs f
 {-# INLINE diff' #-}
 
+-- | Compute a directional derivative of a function @(f a -> a)@
 du :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f (a, a) -> a
 du f = d . getADTower . f . fmap withD
 {-# INLINE du #-}
 
+-- | Compute the answer and a directional derivative of a function @(f a -> a)@
 du' :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f (a, a) -> (a, a)
 du' f = d' . getADTower . f . fmap withD
 {-# INLINE du' #-}
 
+-- | Compute a directional derivative of a function @(f a -> g a)@
 duF :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f (a, a) -> g a
 duF f = fmap (d . getADTower) . f . fmap withD
 {-# INLINE duF #-}
 
+-- | Compute the answer and a directional derivative of a function @(f a -> g a)@
 duF' :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f (a, a) -> g (a, a)
 duF' f = fmap (d' . getADTower) . f . fmap withD
 {-# INLINE duF' #-}
 
+-- | Given a function @(f a -> a)@, and a tower of derivatives, compute the corresponding directional derivatives.
 dus :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f [a] -> [a]
 dus f = getADTower . f . fmap tower
 {-# INLINE dus #-}
 
+-- | Given a function @(f a -> a)@, and a tower of derivatives, compute the corresponding directional derivatives, zero-padded
 dus0 :: (Functor f, Num a) => (f (Tower a) -> Tower a) -> f [a] -> [a]
 dus0 f = zeroPad . getADTower . f . fmap tower
 {-# INLINE dus0 #-}
 
+-- | Given a function @(f a -> g a)@, and a tower of derivatives, compute the corresponding directional derivatives
 dusF :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f [a] -> g [a]
 dusF f = fmap getADTower . f . fmap tower
 {-# INLINE dusF #-}
 
+-- | Given a function @(f a -> g a)@, and a tower of derivatives, compute the corresponding directional derivatives, zero-padded
 dus0F :: (Functor f, Functor g, Num a) => (f (Tower a) -> g (Tower a)) -> f [a] -> g [a]
 dus0F f = fmap getADTower . f . fmap tower
 {-# INLINE dus0F #-}