diff --git a/Project.toml b/Project.toml
index 43f0322..84b1e72 100644
--- a/Project.toml
+++ b/Project.toml
@@ -8,12 +8,8 @@ ArrayLayouts = "4c555306-a7a7-4459-81d9-ec55ddd5c99a"
 BandedMatrices = "aae01518-5342-5314-be14-df237901396f"
 DiffRules = "b552c78f-8df3-52c6-915a-8e097449b14b"
 FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b"
-ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
-Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
-ProfileView = "c46f51b8-102a-5cf2-8d2c-8597cb0e0da7"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
-StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 
 [compat]
 DiffRules = "1.15.1"
@@ -21,9 +17,7 @@ DifferentialEquations = "7.17.0"
 ForwardDiff = "1.2.2"
 Lux = "1.29.1"
 Plots = "1.41.1"
-ProfileView = "1.10.2"
 SparseArrays = "1.10"
-StaticArrays = "1.9.16"
 
 [extras]
 BandedMatrices = "aae01518-5342-5314-be14-df237901396f"
@@ -31,11 +25,9 @@ DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa"
 ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
 Lux = "b2108857-7c20-44ae-9111-449ecde12c47"
 Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
-Revise = "295af30f-e4ad-537b-8983-00126c2a3abe"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-dev = ["Revise"]
 examples = ["Plots", "DifferentialEquations", "ForwardDiff", "BandedMatrices", "Lux"]
 test = ["Test", "ForwardDiff", "BandedMatrices", "SparseArrays"]
diff --git a/src/DualArrays.jl b/src/DualArrays.jl
index 2d30c3d..bbf3392 100644
--- a/src/DualArrays.jl
+++ b/src/DualArrays.jl
@@ -13,7 +13,7 @@ module DualArrays
 
 export DualVector, Dual, jacobian
 
-import Base: +, -, ==, getindex, size, axes, broadcasted, show, sum, vcat, convert, *, isapprox
+import Base: +, -, ==, getindex, size, axes, broadcasted, show, sum, vcat, convert, *, isapprox, promote_type
 
 using LinearAlgebra, ArrayLayouts, FillArrays, DiffRules
 
diff --git a/src/arithmetic.jl b/src/arithmetic.jl
index 998b50b..0749552 100644
--- a/src/arithmetic.jl
+++ b/src/arithmetic.jl
@@ -77,6 +77,16 @@ for (_, f, n) in DiffRules.diffrules(filter_modules=(:Base,))
             jac = $p1.(x.value, y.value) .* x.jacobian .+ $p2.(x.value, y.value) .* y.jacobian
             return DualVector(val, jac)
         end
+        @eval function broadcasted(::typeof($f), x::Dual, y::AbstractVector)
+            val = $f.(x.value, y)
+            jac = $p1.(x.value, y) .* transpose(x.partials)
+            return DualVector(val, jac)
+        end
+        @eval function broadcasted(::typeof($f), x::AbstractVector, y::Dual)
+            val = $f.(x, y.value)
+            jac = $p2.(x, y.value) .* transpose(y.partials)
+            return DualVector(val, jac)
+        end
         # Must have Base.$f in order not to import everything
         @eval Base.$f(x::Dual, y::Dual) = Dual($f(x.value, y.value), $p1(x.value, y.value) * x.partials + $p2(x.value, y.value) * y.partials)
         @eval Base.$f(x::Dual, y::Real) = Dual($f(x.value, y), $p1(x.value, y) * x.partials)
diff --git a/src/types.jl b/src/types.jl
index 5367511..62712fb 100644
--- a/src/types.jl
+++ b/src/types.jl
@@ -10,10 +10,12 @@ A dual number type that stores a value and its partials (derivatives).
 - `partials::Partials`: The partial derivatives as a vector
 """
 struct Dual{T, Partials <: AbstractVector{T}} <: Real
-    value::T
+    value::Union{T, Dual{T}}
     partials::Partials
 end
 
+
+
 """
     DualVector{T, M <: AbstractMatrix{T}} <: AbstractVector{Dual{T}}
 
@@ -32,11 +34,11 @@ For now the entries just return the values when indexed.
 
 Constructs a DualVector, ensuring that the vector length matches the number of rows in the Jacobian.
 """
-struct DualVector{T, V <: AbstractVector{T},M <: AbstractMatrix{T}} <: AbstractVector{Dual{T}}
+struct DualVector{T, V <: AbstractVector{<:Union{T, Dual{T}}},M <: AbstractMatrix{T}} <: AbstractVector{Dual{T}}
     value::V
     jacobian::M
 
-    function DualVector(value::V, jacobian::M) where {T, V <: AbstractVector{T}, M <: AbstractMatrix{T}}
+    function DualVector(value::V, jacobian::M) where {T, V <: AbstractVector{<:Union{T, Dual{T}}}, M <: AbstractMatrix{T}}
         if size(jacobian, 1) != length(value)
             x, y = length(value), size(jacobian, 1)
             throw(ArgumentError("vector length must match number of rows in jacobian.\n" *
@@ -47,12 +49,19 @@ struct DualVector{T, V <: AbstractVector{T},M <: AbstractMatrix{T}} <: AbstractV
     end
 end
 
+promote_type_dual(T, S) = promote_type(T, S)
+
+promote_type_dual(::Type{<:Dual{T}}, S) where {T} = promote_type(T, S)
+promote_type_dual(S, ::Type{<:Dual{T}}) where {T} = promote_type(T, S)
+
+broadcasted(::Type{T}, d::DualVector) where {T} = DualVector(T.(d.value), T.(d.jacobian))
+
 """
 Constructor that forces type compatibility
 """
-function DualVector(value::AbstractVector, jacobian::AbstractMatrix)
-    T = promote_type(eltype(value), eltype(jacobian))
-    DualVector(convert(Vector{T}, value), convert(AbstractMatrix{T}, jacobian))
+function DualVector(value::AbstractVector{T}, jacobian::AbstractMatrix{S}) where {T, S}
+    U = promote_type_dual(T, S)
+    DualVector(U.(value), U.(jacobian))
 end
 
 # Basic equality for Dual numbers
diff --git a/src/utilities.jl b/src/utilities.jl
index 080155f..c656964 100644
--- a/src/utilities.jl
+++ b/src/utilities.jl
@@ -49,8 +49,13 @@ end
 """
 Custom display method for DualVectors.
 """
-Base.show(io::IO, ::MIME"text/plain", x::DualVector) = 
-    (print(io, x.value); print(io, " + "); print(io, x.jacobian); print(io, "𝛜"))
+show_dual_vector(io::IO, x::AbstractArray, ::Any) = print(io, x)
+function show_dual_vector(io::IO, x::DualVector, i = 0)
+    show_dual_vector(io, x.value, i+1)
+    print(io, " + $(x.jacobian)ϵ" * repeat('\'', i))
+end
+Base.show(io::IO, x::DualVector) = show_dual_vector(io, x)
+Base.show(io::IO, ::MIME"text/plain", x::DualVector) = Base.show(io, x)
 
 """
 Utility function to compute the jacobian of a function `f` at point `x`.
diff --git a/test/broadcast_test.jl b/test/broadcast_test.jl
index f87d9f4..7358d9d 100644
--- a/test/broadcast_test.jl
+++ b/test/broadcast_test.jl
@@ -71,4 +71,25 @@ using DualArrays, Test, SparseArrays, LinearAlgebra
     @test s.jacobian ≈ [0.0 -2.0 -3.0; -1.0 -1.0 -3.0; -1.0 -2.0 -2.0]
     @test m.jacobian ≈ [3.0 2.0 3.0; 2.0 6.0 6.0; 3.0 6.0 11.0]
     @test div.jacobian ≈ [0.25 -0.5 -0.75; -0.5 -0.5 -1.5; -0.75 -1.5 -1.75]
+
+    # Broadcasting between Dual and AbstractVector
+    a = x .+ [1.0, 2.0, 3.0]
+    s = [1.0, 2.0, 3.0] .- x
+    m = x .* [1.0, 2.0, 3.0]
+    div = [1.0, 2.0, 3.0] ./ x
+
+    @test a isa DualVector
+    @test s isa DualVector
+    @test m isa DualVector
+    @test div isa DualVector
+
+    @test a.value ≈ [3.0, 4.0, 5.0]
+    @test s.value ≈ [-1.0, 0.0, 1.0]
+    @test m.value ≈ [2.0, 4.0, 6.0]
+    @test div.value ≈ [0.5, 1.0, 1.5]
+
+    @test a.jacobian ≈ [1.0 2.0 3.0; 1.0 2.0 3.0; 1.0 2.0 3.0]
+    @test s.jacobian ≈ [-1.0 -2.0 -3.0; -1.0 -2.0 -3.0; -1.0 -2.0 -3.0]
+    @test m.jacobian ≈ [1.0 2.0 3.0; 2.0 4.0 6.0; 3.0 6.0 9.0]
+    @test div.jacobian ≈ [-0.25 -0.5 -0.75; -0.5 -1.0 -1.5; -0.75 -1.5 -2.25]
 end
\ No newline at end of file
diff --git a/test/runtests.jl b/test/runtests.jl
index 729b4ed..30ccb21 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -81,6 +81,17 @@ using DualArrays: Dual
         @test vcat(x, x) == DualVector([1, 1], [1 2 3;1 2 3])
         @test vcat(x, y) == DualVector([1, 2, 3], [1 2 3;4 5 6;7 8 9])
     end
+
+    @testset "Hessian" begin
+        d = DualVector(
+            DualVector([1, 2], [1 0;0 1]),
+            [1 0;0 1]
+        )
+        f(x) = x[1] * x[2]
+        @test f(d) isa Dual
+        @test f(d).partials isa DualVector
+        @test f(d).partials.jacobian == [0 1; 1 0]
+    end
     
     include("broadcast_test.jl")
 end
\ No newline at end of file