diff --git a/Project.toml b/Project.toml
index e0d9275d..205cf836 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "SpecialFunctions"
 uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
-version = "1.7.0"
+version = "1.8.0"
 
 [deps]
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
diff --git a/src/chainrules.jl b/src/chainrules.jl
index fa7b5dd3..ad79dd03 100644
--- a/src/chainrules.jl
+++ b/src/chainrules.jl
@@ -193,3 +193,93 @@ ChainRulesCore.@scalar_rule(
 ChainRulesCore.@scalar_rule(expinti(x), exp(x) / x)
 ChainRulesCore.@scalar_rule(sinint(x), sinc(invπ * x))
 ChainRulesCore.@scalar_rule(cosint(x), cos(x) / x)
+
+# non-holomorphic functions
+function ChainRulesCore.frule((_, Δν, Δx), ::typeof(besselix), ν::Number, x::Number)
+    # primal
+    Ω = besselix(ν, x)
+
+    # derivative
+    ∂Ω_∂ν = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
+    a = (besselix(ν - 1, x) + besselix(ν + 1, x)) / 2
+    ΔΩ = if Δx isa Real
+        muladd(muladd(-sign(real(x)), Ω, a), Δx, ∂Ω_∂ν * Δν)
+    else
+        muladd(a, Δx, muladd(-sign(real(x)) * real(Δx), Ω, ∂Ω_∂ν * Δν))
+    end
+
+    return Ω, ΔΩ
+end
+function ChainRulesCore.rrule(::typeof(besselix), ν::Number, x::Number)
+    Ω = besselix(ν, x)
+    project_x = ChainRulesCore.ProjectTo(x)
+    function besselix_pullback(ΔΩ)
+        ν̄ = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
+        a = (besselix(ν - 1, x) + besselix(ν + 1, x)) / 2
+        x̄ = project_x(muladd(conj(a), ΔΩ, - sign(real(x)) * real(conj(Ω) * ΔΩ)))
+        return ChainRulesCore.NoTangent(), ν̄, x̄
+    end
+    return Ω, besselix_pullback
+end
+
+function ChainRulesCore.frule((_, Δν, Δx), ::typeof(besseljx), ν::Number, x::Number)
+    # primal
+    Ω = besseljx(ν, x)
+
+    # derivative
+    ∂Ω_∂ν = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
+    a = (besseljx(ν - 1, x) - besseljx(ν + 1, x)) / 2
+    ΔΩ = if Δx isa Real
+        muladd(a, Δx, ∂Ω_∂ν * Δν)
+    else
+        muladd(a, Δx, muladd(-sign(imag(x)) * imag(Δx), Ω, ∂Ω_∂ν * Δν))
+    end
+
+    return Ω, ΔΩ
+end
+function ChainRulesCore.rrule(::typeof(besseljx), ν::Number, x::Number)
+    Ω = besseljx(ν, x)
+    project_x = ChainRulesCore.ProjectTo(x)
+    function besseljx_pullback(ΔΩ)
+        ν̄ = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
+        a = (besseljx(ν - 1, x) - besseljx(ν + 1, x)) / 2
+        x̄ = if x isa Real
+            project_x(a * ΔΩ)
+        else
+            project_x(muladd(conj(a), ΔΩ, - sign(imag(x)) * real(conj(Ω) * ΔΩ) * im))
+        end
+        return ChainRulesCore.NoTangent(), ν̄, x̄
+    end
+    return Ω, besseljx_pullback
+end
+
+function ChainRulesCore.frule((_, Δν, Δx), ::typeof(besselyx), ν::Number, x::Number)
+    # primal
+    Ω = besselyx(ν, x)
+
+    # derivative
+    ∂Ω_∂ν = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
+    a = (besselyx(ν - 1, x) - besselyx(ν + 1, x)) / 2
+    ΔΩ = if Δx isa Real
+        muladd(a, Δx, ∂Ω_∂ν * Δν)
+    else
+        muladd(a, Δx, muladd(-sign(imag(x)) * imag(Δx), Ω, ∂Ω_∂ν * Δν))
+    end
+
+    return Ω, ΔΩ
+end
+function ChainRulesCore.rrule(::typeof(besselyx), ν::Number, x::Number)
+    Ω = besselyx(ν, x)
+    project_x = ChainRulesCore.ProjectTo(x)
+    function besselyx_pullback(ΔΩ)
+        ν̄ = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
+        a = (besselyx(ν - 1, x) - besselyx(ν + 1, x)) / 2
+        x̄ = if x isa Real
+            project_x(a * ΔΩ)
+        else
+            project_x(muladd(conj(a), ΔΩ, - sign(imag(x)) * real(conj(Ω) * ΔΩ) * im))
+        end
+        return ChainRulesCore.NoTangent(), ν̄, x̄
+    end
+    return Ω, besselyx_pullback
+end
diff --git a/test/chainrules.jl b/test/chainrules.jl
index d4be8285..506b4b48 100644
--- a/test/chainrules.jl
+++ b/test/chainrules.jl
@@ -53,9 +53,15 @@
             for nu in (-1.5, 2.2, 4.0)
                 test_frule(besseli, nu, x)
                 test_rrule(besseli, nu, x)
+                test_frule(besselix, nu, x) # derivative is `NotImplemented`
+                test_frule(besselix, nu ⊢ NoTangent(), x) # derivative is a number
+                test_rrule(besselix, nu, x)
 
                 test_frule(besselj, nu, x)
                 test_rrule(besselj, nu, x)
+                test_frule(besseljx, nu, x) # derivative is `NotImplemented`
+                test_frule(besseljx, nu ⊢ NoTangent(), x) # derivative is a number
+                test_rrule(besseljx, nu, x)
 
                 test_frule(besselk, nu, x)
                 test_rrule(besselk, nu, x)
@@ -64,6 +70,9 @@
 
                 test_frule(bessely, nu, x)
                 test_rrule(bessely, nu, x)
+                test_frule(besselyx, nu, x) # derivative is `NotImplemented`
+                test_frule(besselyx, nu ⊢ NoTangent(), x) # derivative is a number
+                test_rrule(besselyx, nu, x)
 
                 test_frule(hankelh1, nu, x)
                 test_rrule(hankelh1, nu, x)