diff --git a/ext/ForwardDiffStaticArraysExt.jl b/ext/ForwardDiffStaticArraysExt.jl
index cb9222ff..1ec87ea7 100644
--- a/ext/ForwardDiffStaticArraysExt.jl
+++ b/ext/ForwardDiffStaticArraysExt.jl
@@ -37,13 +37,13 @@ function LinearAlgebra.eigen(A::Symmetric{<:Dual{Tg,T,N}, <:StaticArrays.StaticM
 end
 
 # Gradient
-@inline ForwardDiff.gradient(f, x::StaticArray)                      = vector_mode_gradient(f, x)
-@inline ForwardDiff.gradient(f, x::StaticArray, cfg::GradientConfig) = gradient(f, x)
-@inline ForwardDiff.gradient(f, x::StaticArray, cfg::GradientConfig, ::Val) = gradient(f, x)
+@inline ForwardDiff.gradient(f::F, x::StaticArray) where F                      = vector_mode_gradient(f, x)
+@inline ForwardDiff.gradient(f::F, x::StaticArray, cfg::GradientConfig) where F = gradient(f, x)
+@inline ForwardDiff.gradient(f::F, x::StaticArray, cfg::GradientConfig, ::Val) where F = gradient(f, x)
 
-@inline ForwardDiff.gradient!(result::Union{AbstractArray,DiffResult}, f, x::StaticArray) = vector_mode_gradient!(result, f, x)
-@inline ForwardDiff.gradient!(result::Union{AbstractArray,DiffResult}, f, x::StaticArray, cfg::GradientConfig) = gradient!(result, f, x)
-@inline ForwardDiff.gradient!(result::Union{AbstractArray,DiffResult}, f, x::StaticArray, cfg::GradientConfig, ::Val) = gradient!(result, f, x)
+@inline ForwardDiff.gradient!(result::Union{AbstractArray,DiffResult}, f::F, x::StaticArray) where F = vector_mode_gradient!(result, f, x)
+@inline ForwardDiff.gradient!(result::Union{AbstractArray,DiffResult}, f::F, x::StaticArray, cfg::GradientConfig) where F = gradient!(result, f, x)
+@inline ForwardDiff.gradient!(result::Union{AbstractArray,DiffResult}, f::F, x::StaticArray, cfg::GradientConfig, ::Val) where F = gradient!(result, f, x)
 
 @generated function extract_gradient(::Type{T}, y::Real, x::S) where {T,S<:StaticArray}
     result = Expr(:tuple, [:(partials(T, y, $i)) for i in 1:length(x)]...)
@@ -65,13 +65,13 @@ end
 end
 
 # Jacobian
-@inline ForwardDiff.jacobian(f, x::StaticArray) = vector_mode_jacobian(f, x)
-@inline ForwardDiff.jacobian(f, x::StaticArray, cfg::JacobianConfig) = jacobian(f, x)
-@inline ForwardDiff.jacobian(f, x::StaticArray, cfg::JacobianConfig, ::Val) = jacobian(f, x)
+@inline ForwardDiff.jacobian(f::F, x::StaticArray) where F = vector_mode_jacobian(f, x)
+@inline ForwardDiff.jacobian(f::F, x::StaticArray, cfg::JacobianConfig) where F  = jacobian(f, x)
+@inline ForwardDiff.jacobian(f::F, x::StaticArray, cfg::JacobianConfig, ::Val)  where F = jacobian(f, x)
 
-@inline ForwardDiff.jacobian!(result::Union{AbstractArray,DiffResult}, f, x::StaticArray) = vector_mode_jacobian!(result, f, x)
-@inline ForwardDiff.jacobian!(result::Union{AbstractArray,DiffResult}, f, x::StaticArray, cfg::JacobianConfig) = jacobian!(result, f, x)
-@inline ForwardDiff.jacobian!(result::Union{AbstractArray,DiffResult}, f, x::StaticArray, cfg::JacobianConfig, ::Val) = jacobian!(result, f, x)
+@inline ForwardDiff.jacobian!(result::Union{AbstractArray,DiffResult}, f::F, x::StaticArray)  where F = vector_mode_jacobian!(result, f, x)
+@inline ForwardDiff.jacobian!(result::Union{AbstractArray,DiffResult}, f::F, x::StaticArray, cfg::JacobianConfig) where F  = jacobian!(result, f, x)
+@inline ForwardDiff.jacobian!(result::Union{AbstractArray,DiffResult}, f::F, x::StaticArray, cfg::JacobianConfig, ::Val) where F  = jacobian!(result, f, x)
 
 @generated function extract_jacobian(::Type{T}, ydual::StaticArray, x::S) where {T,S<:StaticArray}
     M, N = length(ydual), length(x)
@@ -110,18 +110,18 @@ end
 end
 
 # Hessian
-ForwardDiff.hessian(f, x::StaticArray) = jacobian(y -> gradient(f, y), x)
-ForwardDiff.hessian(f, x::StaticArray, cfg::HessianConfig) = hessian(f, x)
-ForwardDiff.hessian(f, x::StaticArray, cfg::HessianConfig, ::Val) = hessian(f, x)
+ForwardDiff.hessian(f::F, x::StaticArray) where F = jacobian(y -> gradient(f, y), x)
+ForwardDiff.hessian(f::F, x::StaticArray, cfg::HessianConfig) where F = hessian(f, x)
+ForwardDiff.hessian(f::F, x::StaticArray, cfg::HessianConfig, ::Val) where F = hessian(f, x)
 
-ForwardDiff.hessian!(result::AbstractArray, f, x::StaticArray) = jacobian!(result, y -> gradient(f, y), x)
+ForwardDiff.hessian!(result::AbstractArray, f::F, x::StaticArray) where F = jacobian!(result, y -> gradient(f, y), x)
 
-ForwardDiff.hessian!(result::MutableDiffResult, f, x::StaticArray) = hessian!(result, f, x, HessianConfig(f, result, x))
+ForwardDiff.hessian!(result::MutableDiffResult, f::F, x::StaticArray) where F = hessian!(result, f, x, HessianConfig(f, result, x))
 
-ForwardDiff.hessian!(result::ImmutableDiffResult, f, x::StaticArray, cfg::HessianConfig) = hessian!(result, f, x)
-ForwardDiff.hessian!(result::ImmutableDiffResult, f, x::StaticArray, cfg::HessianConfig, ::Val) = hessian!(result, f, x)
+ForwardDiff.hessian!(result::ImmutableDiffResult, f::F, x::StaticArray, cfg::HessianConfig) where F = hessian!(result, f, x)
+ForwardDiff.hessian!(result::ImmutableDiffResult, f::F, x::StaticArray, cfg::HessianConfig, ::Val) where F = hessian!(result, f, x)
 
-function ForwardDiff.hessian!(result::ImmutableDiffResult, f, x::StaticArray)
+function ForwardDiff.hessian!(result::ImmutableDiffResult, f::F, x::StaticArray) where F
     T = typeof(Tag(f, eltype(x)))
     d1 = dualize(T, x)
     d2 = dualize(T, d1)
diff --git a/src/derivative.jl b/src/derivative.jl
index accc2405..b39e2a48 100644
--- a/src/derivative.jl
+++ b/src/derivative.jl
@@ -22,8 +22,8 @@ stored in `y`.
 
 Set `check` to `Val{false}()` to disable tag checking. This can lead to perturbation confusion, so should be used with care.
 """
-@inline function derivative(f!, y::AbstractArray, x::Real,
-                            cfg::DerivativeConfig{T} = DerivativeConfig(f!, y, x), ::Val{CHK}=Val{true}()) where {T, CHK}
+@inline function derivative(f!::F, y::AbstractArray, x::Real,
+                            cfg::DerivativeConfig{T} = DerivativeConfig(f!, y, x), ::Val{CHK}=Val{true}()) where {F, T, CHK}
     require_one_based_indexing(y)
     CHK && checktag(T, f!, x)
     ydual = cfg.duals
@@ -60,8 +60,8 @@ called as `f!(y, x)` where the result is stored in `y`.
 Set `check` to `Val{false}()` to disable tag checking. This can lead to perturbation confusion, so should be used with care.
 """
 @inline function derivative!(result::Union{AbstractArray,DiffResult},
-                             f!, y::AbstractArray, x::Real,
-                             cfg::DerivativeConfig{T} = DerivativeConfig(f!, y, x), ::Val{CHK}=Val{true}()) where {T, CHK}
+                             f!::F, y::AbstractArray, x::Real,
+                             cfg::DerivativeConfig{T} = DerivativeConfig(f!, y, x), ::Val{CHK}=Val{true}()) where {F, T, CHK}
     result isa DiffResult ? require_one_based_indexing(y) : require_one_based_indexing(result, y)
     CHK && checktag(T, f!, x)
     ydual = cfg.duals
diff --git a/src/gradient.jl b/src/gradient.jl
index c0b4d709..6d4cc791 100644
--- a/src/gradient.jl
+++ b/src/gradient.jl
@@ -13,7 +13,7 @@ This method assumes that `isa(f(x), Real)`.
 
 Set `check` to `Val{false}()` to disable tag checking. This can lead to perturbation confusion, so should be used with care.
 """
-function gradient(f, x::AbstractArray, cfg::GradientConfig{T} = GradientConfig(f, x), ::Val{CHK}=Val{true}()) where {T, CHK}
+function gradient(f::F, x::AbstractArray, cfg::GradientConfig{T} = GradientConfig(f, x), ::Val{CHK}=Val{true}()) where {F, T, CHK}
     require_one_based_indexing(x)
     CHK && checktag(T, f, x)
     if chunksize(cfg) == length(x)
diff --git a/src/hessian.jl b/src/hessian.jl
index ababac0f..9c755c9a 100644
--- a/src/hessian.jl
+++ b/src/hessian.jl
@@ -11,7 +11,7 @@ This method assumes that `isa(f(x), Real)`.
 
 Set `check` to `Val{false}()` to disable tag checking. This can lead to perturbation confusion, so should be used with care.
 """
-function hessian(f, x::AbstractArray, cfg::HessianConfig{T} = HessianConfig(f, x), ::Val{CHK}=Val{true}()) where {T,CHK}
+function hessian(f::F, x::AbstractArray, cfg::HessianConfig{T} = HessianConfig(f, x), ::Val{CHK}=Val{true}()) where {F, T,CHK}
     require_one_based_indexing(x)
     CHK && checktag(T, f, x)
     ∇f = y -> gradient(f, y, cfg.gradient_config, Val{false}())
@@ -28,7 +28,7 @@ This method assumes that `isa(f(x), Real)`.
 
 Set `check` to `Val{false}()` to disable tag checking. This can lead to perturbation confusion, so should be used with care.
 """
-function hessian!(result::AbstractArray, f, x::AbstractArray, cfg::HessianConfig{T} = HessianConfig(f, x), ::Val{CHK}=Val{true}()) where {T,CHK}
+function hessian!(result::AbstractArray, f::F, x::AbstractArray, cfg::HessianConfig{T} = HessianConfig(f, x), ::Val{CHK}=Val{true}()) where {F,T,CHK}
     require_one_based_indexing(result, x)
     CHK && checktag(T, f, x)
     ∇f = y -> gradient(f, y, cfg.gradient_config, Val{false}())
@@ -63,8 +63,7 @@ because `isa(result, DiffResult)`, `cfg` is constructed as `HessianConfig(f, res
 
 Set `check` to `Val{false}()` to disable tag checking. This can lead to perturbation confusion, so should be used with care.
 """
-function hessian!(result::DiffResult, f, x::AbstractArray, cfg::HessianConfig{T} = HessianConfig(f, result, x), ::Val{CHK}=Val{true}()) where {T,CHK}
-    require_one_based_indexing(x)
+function hessian!(result::DiffResult, f::F, x::AbstractArray, cfg::HessianConfig{T} = HessianConfig(f, result, x), ::Val{CHK}=Val{true}()) where {F,T,CHK}
     CHK && checktag(T, f, x)
     ∇f! = InnerGradientForHess(result, cfg, f)
     jacobian!(DiffResults.hessian(result), ∇f!, DiffResults.gradient(result), x, cfg.jacobian_config, Val{false}())
diff --git a/src/jacobian.jl b/src/jacobian.jl
index 1ed3d7b5..e7ca96f5 100644
--- a/src/jacobian.jl
+++ b/src/jacobian.jl
@@ -15,7 +15,7 @@ This method assumes that `isa(f(x), AbstractArray)`.
 
 Set `check` to `Val{false}()` to disable tag checking. This can lead to perturbation confusion, so should be used with care.
 """
-function jacobian(f, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f, x), ::Val{CHK}=Val{true}()) where {T,CHK}
+function jacobian(f::F, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f, x), ::Val{CHK}=Val{true}()) where {F,T,CHK}
     require_one_based_indexing(x)
     CHK && checktag(T, f, x)
     if chunksize(cfg) == length(x)
@@ -33,7 +33,7 @@ stored in `y`.
 
 Set `check` to `Val{false}()` to disable tag checking. This can lead to perturbation confusion, so should be used with care.
 """
-function jacobian(f!, y::AbstractArray, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f!, y, x), ::Val{CHK}=Val{true}()) where {T, CHK}
+function jacobian(f!::F, y::AbstractArray, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f!, y, x), ::Val{CHK}=Val{true}()) where {F,T, CHK}
     require_one_based_indexing(y, x)
     CHK && checktag(T, f!, x)
     if chunksize(cfg) == length(x)
@@ -54,7 +54,7 @@ This method assumes that `isa(f(x), AbstractArray)`.
 
 Set `check` to `Val{false}()` to disable tag checking. This can lead to perturbation confusion, so should be used with care.
 """
-function jacobian!(result::Union{AbstractArray,DiffResult}, f, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f, x), ::Val{CHK}=Val{true}()) where {T, CHK}
+function jacobian!(result::Union{AbstractArray,DiffResult}, f::F, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f, x), ::Val{CHK}=Val{true}()) where {F,T, CHK}
     result isa DiffResult ? require_one_based_indexing(x) : require_one_based_indexing(result, x)
     CHK && checktag(T, f, x)
     if chunksize(cfg) == length(x)
@@ -75,7 +75,7 @@ This method assumes that `isa(f(x), AbstractArray)`.
 
 Set `check` to `Val{false}()` to disable tag checking. This can lead to perturbation confusion, so should be used with care.
 """
-function jacobian!(result::Union{AbstractArray,DiffResult}, f!, y::AbstractArray, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f!, y, x), ::Val{CHK}=Val{true}()) where {T,CHK}
+function jacobian!(result::Union{AbstractArray,DiffResult}, f!::F, y::AbstractArray, x::AbstractArray, cfg::JacobianConfig{T} = JacobianConfig(f!, y, x), ::Val{CHK}=Val{true}()) where {F,T,CHK}
     result isa DiffResult ? require_one_based_indexing(y, x) : require_one_based_indexing(result, y, x)
     CHK && checktag(T, f!, x)
     if chunksize(cfg) == length(x)