diff --git a/Project.toml b/Project.toml
index 83f1256..13d160c 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "DiffRules"
 uuid = "b552c78f-8df3-52c6-915a-8e097449b14b"
-version = "1.11.1"
+version = "1.12.0"
 
 [deps]
 IrrationalConstants = "92d709cd-6900-40b7-9082-c6be49f344b6"
diff --git a/src/rules.jl b/src/rules.jl
index 6e4a1fa..03df753 100644
--- a/src/rules.jl
+++ b/src/rules.jl
@@ -116,6 +116,9 @@ _abs_deriv(x) = signbit(x) ? -one(x) : one(x)
 @define_diffrule SpecialFunctions.erfinv(x)      =
     :(  ($sqrtπ * exp(SpecialFunctions.erfinv($x)^2)) / 2  )
 @define_diffrule SpecialFunctions.erfc(x)        = :( -($invsqrtπ * exp(-$x^2) * 2)       )
+@define_diffrule SpecialFunctions.logerfc(x)     =
+    :( - 2 * ($invsqrtπ * exp(- $x^2 - SpecialFunctions.logerfc($x))) )
+
 @define_diffrule SpecialFunctions.erfcinv(x)     =
     :( -($sqrtπ * exp(SpecialFunctions.erfcinv($x)^2)) / 2  )
 @define_diffrule SpecialFunctions.erfi(x)        = :(  $invsqrtπ * exp($x^2) * 2        )
@@ -123,6 +126,7 @@ _abs_deriv(x) = signbit(x) ? -one(x) : one(x)
     :(  2 * (($x * SpecialFunctions.erfcx($x)) - $invsqrtπ)  )
 @define_diffrule SpecialFunctions.logerfcx(x) =
     :(  2 * ($x - inv(SpecialFunctions.erfcx($x) * $sqrtπ))  )
+
 @define_diffrule SpecialFunctions.dawson(x)      =
     :(  1 - (2 * $x * SpecialFunctions.dawson($x))  )
 @define_diffrule SpecialFunctions.digamma(x) =
@@ -131,14 +135,35 @@ _abs_deriv(x) = signbit(x) ? -one(x) : one(x)
     :(  inv(SpecialFunctions.trigamma(SpecialFunctions.invdigamma($x)))  )
 @define_diffrule SpecialFunctions.trigamma(x)    =
     :(  SpecialFunctions.polygamma(2, $x)  )
+
+# derivatives for `airybix` and `airybiprimex` are only correct for real inputs
+# `airyaix` and `airyaiprimex` are only defined for positive real inputs
+# `airybix` and `airybiprimex` are unscaled for negative real inputs
 @define_diffrule SpecialFunctions.airyai(x)      =
     :(  SpecialFunctions.airyaiprime($x)  )
 @define_diffrule SpecialFunctions.airyaiprime(x) =
     :(  $x * SpecialFunctions.airyai($x)  )
+@define_diffrule SpecialFunctions.airyaix(x)     =
+    :(  SpecialFunctions.airyaiprimex($x) + sqrt($x) * SpecialFunctions.airyaix($x)  )
+@define_diffrule SpecialFunctions.airyaiprimex(x) =
+    :(  $x * SpecialFunctions.airyaix($x) + sqrt($x) * SpecialFunctions.airyaiprimex($x)  )
 @define_diffrule SpecialFunctions.airybi(x)      =
     :(  SpecialFunctions.airybiprime($x)  )
 @define_diffrule SpecialFunctions.airybiprime(x) =
     :(  $x * SpecialFunctions.airybi($x)  )
+@define_diffrule SpecialFunctions.airybix(x)     =
+    :(  if $x > zero($x)
+            SpecialFunctions.airybiprimex($x) - sqrt($x) * SpecialFunctions.airybix($x)
+        else
+            SpecialFunctions.airybiprimex($x)
+        end  )
+@define_diffrule SpecialFunctions.airybiprimex(x) =
+    :(  if $x > zero($x)
+            $x * SpecialFunctions.airybix($x) - sqrt($x) * SpecialFunctions.airybiprimex($x)
+        else
+            $x * SpecialFunctions.airybix($x)
+        end  )
+
 @define_diffrule SpecialFunctions.besselj0(x)    =
     :( -SpecialFunctions.besselj1($x)  )
 @define_diffrule SpecialFunctions.besselj1(x)    =
@@ -148,49 +173,72 @@ _abs_deriv(x) = signbit(x) ? -one(x) : one(x)
 @define_diffrule SpecialFunctions.bessely1(x)    =
     :(  (SpecialFunctions.bessely0($x) - SpecialFunctions.bessely(2, $x)) / 2 )
 
+@define_diffrule SpecialFunctions.sinint(x) = :( sinc($x / π) )
+@define_diffrule SpecialFunctions.cosint(x) = :( cos($x) / $x )
+
+@define_diffrule SpecialFunctions.ellipk(m) =
+    :( (SpecialFunctions.ellipe($m) / (1 - $m) - SpecialFunctions.ellipk($m)) / (2 * $m) )
+@define_diffrule SpecialFunctions.ellipe(m) =
+    :( (SpecialFunctions.ellipe($m) - SpecialFunctions.ellipk($m)) / (2 * $m) )
+
 # TODO:
 #
 # eta
 # zeta
-# airyaix
-# airyaiprimex
-# airybix
-# airybiprimex
 
 # binary #
 #--------#
 
+@define_diffrule SpecialFunctions.erf(x, y) =
+    :(  -2 * ($invsqrtπ * exp(-$x^2))  ), :(  2 * ($invsqrtπ * exp(-$y^2))  )
+
+# derivatives with respect to the order `ν` exist but are not implemented
+# (analogously to the ChainRules definitions in SpecialFunctions)
+
+# derivatives for `besselix`, `besseljx` and `besselyx` are only correct for real inputs
+# see https://github.com/JuliaMath/SpecialFunctions.jl/blob/master/src/chainrules.jl
+# for forward-mode and reverse-mode derivatives for complex inputs
+
 @define_diffrule SpecialFunctions.besselj(ν, x)   =
     :NaN, :(  (SpecialFunctions.besselj($ν - 1, $x) - SpecialFunctions.besselj($ν + 1, $x)) / 2  )
+@define_diffrule SpecialFunctions.besseljx(ν, x)  =
+    :NaN, :(  (SpecialFunctions.besseljx($ν - 1, $x) - SpecialFunctions.besseljx($ν + 1, $x)) / 2  )
 @define_diffrule SpecialFunctions.besseli(ν, x)   =
     :NaN, :(  (SpecialFunctions.besseli($ν - 1, $x) + SpecialFunctions.besseli($ν + 1, $x)) / 2  )
+@define_diffrule SpecialFunctions.besselix(ν, x)  =
+    :NaN, :(  (SpecialFunctions.besselix($ν - 1, $x) + SpecialFunctions.besselix($ν + 1, $x)) / 2 - sign($x) * SpecialFunctions.besselix($ν, $x)  )
 @define_diffrule SpecialFunctions.bessely(ν, x)   =
     :NaN, :(  (SpecialFunctions.bessely($ν - 1, $x) - SpecialFunctions.bessely($ν + 1, $x)) / 2  )
+@define_diffrule SpecialFunctions.besselyx(ν, x)  =
+    :NaN, :(  (SpecialFunctions.besselyx($ν - 1, $x) - SpecialFunctions.besselyx($ν + 1, $x)) / 2  )
 @define_diffrule SpecialFunctions.besselk(ν, x)   =
     :NaN, :( -(SpecialFunctions.besselk($ν - 1, $x) + SpecialFunctions.besselk($ν + 1, $x)) / 2  )
+@define_diffrule SpecialFunctions.besselkx(ν, x)  =
+    :NaN, :( -(SpecialFunctions.besselkx($ν - 1, $x) + SpecialFunctions.besselkx($ν + 1, $x)) / 2 + SpecialFunctions.besselkx($ν, $x)  )
+@define_diffrule SpecialFunctions.besselh(ν, x)   =
+    :NaN, :(  (SpecialFunctions.besselh($ν - 1, $x) - SpecialFunctions.besselh($ν + 1, $x)) / 2  )
+@define_diffrule SpecialFunctions.besselhx(ν, x) =
+    :NaN, :(  (SpecialFunctions.besselhx($ν - 1, $x) - SpecialFunctions.besselhx($ν + 1, $x)) / 2 - im * SpecialFunctions.besselhx($ν, $x)  )
 @define_diffrule SpecialFunctions.hankelh1(ν, x)  =
     :NaN, :(  (SpecialFunctions.hankelh1($ν - 1, $x) - SpecialFunctions.hankelh1($ν + 1, $x)) / 2  )
+@define_diffrule SpecialFunctions.hankelh1x(ν, x) =
+    :NaN, :(  (SpecialFunctions.hankelh1x($ν - 1, $x) - SpecialFunctions.hankelh1x($ν + 1, $x)) / 2 - im * SpecialFunctions.hankelh1x($ν, $x)  )
 @define_diffrule SpecialFunctions.hankelh2(ν, x)  =
     :NaN, :(  (SpecialFunctions.hankelh2($ν - 1, $x) - SpecialFunctions.hankelh2($ν + 1, $x)) / 2  )
+@define_diffrule SpecialFunctions.hankelh2x(ν, x) =
+    :NaN, :(  (SpecialFunctions.hankelh2x($ν - 1, $x) - SpecialFunctions.hankelh2x($ν + 1, $x)) / 2 + im * SpecialFunctions.hankelh2x($ν, $x)  )
+
 @define_diffrule SpecialFunctions.polygamma(m, x) =
     :NaN, :(  SpecialFunctions.polygamma($m + 1, $x)  )
+
 @define_diffrule SpecialFunctions.beta(a, b)      =
     :( SpecialFunctions.beta($a, $b)*(SpecialFunctions.digamma($a) - SpecialFunctions.digamma($a + $b)) ), :(  SpecialFunctions.beta($a, $b)*(SpecialFunctions.digamma($b) - SpecialFunctions.digamma($a + $b))     )
-@define_diffrule SpecialFunctions.logbeta(a, b)     =
+@define_diffrule SpecialFunctions.logbeta(a, b)   =
     :( SpecialFunctions.digamma($a) - SpecialFunctions.digamma($a + $b)  ), :(  SpecialFunctions.digamma($b) - SpecialFunctions.digamma($a + $b)  )
 
-# TODO:
-#
-# zeta
-# besseljx
-# besselyx
-# besselix
-# besselkx
-# besselh
-# besselhx
-# hankelh1x
-# hankelh2
-# hankelh2x
+# derivative wrt to `s` is not implemented
+@define_diffrule SpecialFunctions.zeta(s, z)      =
+    :NaN, :( - $s * SpecialFunctions.zeta($s + 1, $z)  )
 
 # ternary #
 #---------#
diff --git a/test/runtests.jl b/test/runtests.jl
index 3312f7f..f1fa041 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -118,7 +118,7 @@ for xtype in [:Float64, :BigFloat, :Int64]
                 x = $xtype(rand(1 : 10))
                 y = $mode
                 dx, dy = $(derivs[1]), $(derivs[2])
-                @test isapprox(dx, finitediff(z -> rem2pi(z, y), float(x)), rtol=0.05)
+                @test dx ≈ finitediff(z -> rem2pi(z, y), float(x)) rtol=1e-9 atol=1e-9
                 @test isnan(dy)
             end
         end
@@ -134,13 +134,24 @@ for xtype in [:Float64, :BigFloat]
                 x = rand($xtype)
                 y = $ytype(rand(1 : 10))
                 dx, dy = $(derivs[1]), $(derivs[2])
-                @test isapprox(dx, finitediff(z -> ldexp(z, y), x), rtol=0.05)
+                @test dx ≈ finitediff(z -> ldexp(z, y), x) rtol=1e-9 atol=1e-9
                 @test isnan(dy)
             end
         end
     end
 end
 
+# Check negative branch for `airybix` and `airybiprimex`
+for f in (:airybix, :airybiprimex)
+    deriv = DiffRules.diffrule(:SpecialFunctions, f, :goo)
+    @eval begin
+        let
+            goo = -rand()
+            @test $deriv ≈ finitediff(SpecialFunctions.$f, goo) rtol=1e-9 atol=1e-9
+        end
+    end
+end
+
 # Test `iszero(x)` branch of `xlogy`
 derivs = DiffRules.diffrule(:LogExpFunctions, :xlogy, :x, :y)
 for xytype in [:Float32, :Float64, :BigFloat]