diff --git a/src/_ChainRules.jl b/src/_ChainRules.jl
index 81501148e..0470dbab6 100644
--- a/src/_ChainRules.jl
+++ b/src/_ChainRules.jl
@@ -2,15 +2,15 @@ const BSPLINESPACE_INFO = """
 derivatives of B-spline basis functions with respect to BSplineSpace not implemented currently.
 """
 
-function ChainRulesCore.frule((_, ΔP, Δi, Δt), ::typeof(bsplinebasis), P::BSplineSpace, i::Integer, t::Real)
+# bsplinebasis
+function ChainRulesCore.frule((_, ΔP, Δi, Δt), ::typeof(bsplinebasis), P::AbstractFunctionSpace, i::Integer, t::Real)
     B = bsplinebasis(P,i,t)
     ∂B_∂P = @not_implemented BSPLINESPACE_INFO
     # ∂B_∂i = NoTangent()
     ∂B_∂t = bsplinebasis′(P,i,t)
     return (B, ∂B_∂P*ΔP + ∂B_∂t*Δt)
 end
-
-function ChainRulesCore.rrule(::typeof(bsplinebasis), P::BSplineSpace, i::Integer, t::Real)
+function ChainRulesCore.rrule(::typeof(bsplinebasis), P::AbstractFunctionSpace, i::Integer, t::Real)
     B = bsplinebasis(P,i,t)
     # project_t = ProjectTo(t)  # Not sure we need this ProjectTo.
     function bsplinebasis_pullback(ΔB)
@@ -22,15 +22,15 @@ function ChainRulesCore.rrule(::typeof(bsplinebasis), P::BSplineSpace, i::Intege
     return (B, bsplinebasis_pullback)
 end
 
-function ChainRulesCore.frule((_, ΔP, Δi, Δt), ::typeof(bsplinebasis₊₀), P::BSplineSpace, i::Integer, t::Real)
+# bsplinebasis₊₀
+function ChainRulesCore.frule((_, ΔP, Δi, Δt), ::typeof(bsplinebasis₊₀), P::AbstractFunctionSpace, i::Integer, t::Real)
     B = bsplinebasis₊₀(P,i,t)
     ∂B_∂P = @not_implemented BSPLINESPACE_INFO
     # ∂B_∂i = NoTangent()
     ∂B_∂t = bsplinebasis′₊₀(P,i,t)
     return (B, ∂B_∂P*ΔP + ∂B_∂t*Δt)
 end
-
-function ChainRulesCore.rrule(::typeof(bsplinebasis₊₀), P::BSplineSpace, i::Integer, t::Real)
+function ChainRulesCore.rrule(::typeof(bsplinebasis₊₀), P::AbstractFunctionSpace, i::Integer, t::Real)
     B = bsplinebasis₊₀(P,i,t)
     # project_t = ProjectTo(t)  # Not sure we need this ProjectTo.
     function bsplinebasis_pullback(ΔB)
@@ -42,15 +42,15 @@ function ChainRulesCore.rrule(::typeof(bsplinebasis₊₀), P::BSplineSpace, i::
     return (B, bsplinebasis_pullback)
 end
 
-function ChainRulesCore.frule((_, ΔP, Δi, Δt), ::typeof(bsplinebasis₋₀), P::BSplineSpace, i::Integer, t::Real)
+# bsplinebasis₋₀
+function ChainRulesCore.frule((_, ΔP, Δi, Δt), ::typeof(bsplinebasis₋₀), P::AbstractFunctionSpace, i::Integer, t::Real)
     B = bsplinebasis₋₀(P,i,t)
     ∂B_∂P = @not_implemented BSPLINESPACE_INFO
     # ∂B_∂i = NoTangent()
     ∂B_∂t = bsplinebasis′₋₀(P,i,t)
     return (B, ∂B_∂P*ΔP + ∂B_∂t*Δt)
 end
-
-function ChainRulesCore.rrule(::typeof(bsplinebasis₋₀), P::BSplineSpace, i::Integer, t::Real)
+function ChainRulesCore.rrule(::typeof(bsplinebasis₋₀), P::AbstractFunctionSpace, i::Integer, t::Real)
     B = bsplinebasis₋₀(P,i,t)
     # project_t = ProjectTo(t)  # Not sure we need this ProjectTo.
     function bsplinebasis_pullback(ΔB)
@@ -61,3 +61,23 @@ function ChainRulesCore.rrule(::typeof(bsplinebasis₋₀), P::BSplineSpace, i::
     end
     return (B, bsplinebasis_pullback)
 end
+
+# bsplinebasisall
+function ChainRulesCore.frule((_, ΔP, Δi, Δt), ::typeof(bsplinebasisall), P::AbstractFunctionSpace, i::Integer, t::Real)
+    B = bsplinebasisall(P,i,t)
+    ∂B_∂P = @not_implemented BSPLINESPACE_INFO
+    # ∂B_∂i = NoTangent()
+    ∂B_∂t = bsplinebasisall(derivative(P),i,t)
+    return (B, ∂B_∂P*ΔP + ∂B_∂t*Δt)
+end
+function ChainRulesCore.rrule(::typeof(bsplinebasisall), P::AbstractFunctionSpace, i::Integer, t::Real)
+    B = bsplinebasisall(P,i,t)
+    # project_t = ProjectTo(t)  # Not sure we need this ProjectTo.
+    function bsplinebasis_pullback(ΔB)
+        P̄ = @not_implemented BSPLINESPACE_INFO
+        ī = NoTangent()
+        t̄ = bsplinebasisall(derivative(P),i,t)' * ΔB
+        return (NoTangent(), P̄, ī, t̄)
+    end
+    return (B, bsplinebasis_pullback)
+end
diff --git a/src/_DerivativeSpace.jl b/src/_DerivativeSpace.jl
index bcdd1c01e..5fc77f292 100644
--- a/src/_DerivativeSpace.jl
+++ b/src/_DerivativeSpace.jl
@@ -40,6 +40,8 @@ exactdim(dP::BSplineDerivativeSpace{r,<:AbstractBSplineSpace{p}}) where {r,p} =
 intervalindex(dP::BSplineDerivativeSpace,t::Real) = intervalindex(bsplinespace(dP),t)
 domain(dP::BSplineDerivativeSpace) = domain(bsplinespace(dP))
 _lower(dP::BSplineDerivativeSpace{r}) where r = BSplineDerivativeSpace{r-1}(_lower(bsplinespace(dP)))
+derivative(P::BSplineSpace) = BSplineDerivativeSpace{1}(P)
+derivative(dP::BSplineDerivativeSpace{r}) where r = BSplineDerivativeSpace{r+1}(bsplinespace(dP))
 
 function Base.issubset(dP::BSplineDerivativeSpace{r,<:AbstractBSplineSpace{p}}, P′::AbstractBSplineSpace) where {r,p}
     k = knotvector(dP)
diff --git a/test/test_ChainRules.jl b/test/test_ChainRules.jl
index e50d3f81a..8956a5c2d 100644
--- a/test/test_ChainRules.jl
+++ b/test/test_ChainRules.jl
@@ -1,17 +1,26 @@
 @testset "ChainRules" begin
-    k = KnotVector(rand(12))
-    P = BSplineSpace{2}(k)
+    k = KnotVector(rand(20))
+    p = 3
+    P = BSplineSpace{p}(k)
+    dP0 = BSplineDerivativeSpace{0}(P)
+    dP1 = BSplineDerivativeSpace{1}(P)
+    dP2 = BSplineDerivativeSpace{2}(P)
     @testset "bsplinebasis" begin
-        for _ in 1:10
+        for _P in (P, dP0, dP1, dP2), i in 1:dim(_P)
             t = rand(domain(P))
-            for i in 1:dim(P)
-                test_frule(bsplinebasis, P, i, t)
-                test_rrule(bsplinebasis, P, i, t)
-                test_frule(bsplinebasis₊₀, P, i, t)
-                test_rrule(bsplinebasis₊₀, P, i, t)
-                test_frule(bsplinebasis₋₀, P, i, t)
-                test_rrule(bsplinebasis₋₀, P, i, t)
-            end
+            test_frule(bsplinebasis, _P, i, t)
+            test_rrule(bsplinebasis, _P, i, t)
+            test_frule(bsplinebasis₊₀, _P, i, t)
+            test_rrule(bsplinebasis₊₀, _P, i, t)
+            test_frule(bsplinebasis₋₀, _P, i, t)
+            test_rrule(bsplinebasis₋₀, _P, i, t)
+        end
+    end
+    @testset "bsplinebasisall" begin
+        for _P in (P, dP0, dP1, dP2), i in 1:length(k)-2p-1
+            t = rand(domain(P))
+            test_frule(bsplinebasisall, _P, i, t)
+            test_rrule(bsplinebasisall, _P, i, t)
         end
     end
 end
diff --git a/test/test_Derivative.jl b/test/test_Derivative.jl
index ddc08d2f2..2e64bfa4e 100644
--- a/test/test_Derivative.jl
+++ b/test/test_Derivative.jl
@@ -88,8 +88,11 @@
         ts = rand(10)
         for p in 0:5
             P = BSplineSpace{p}(k)
+            Q = P
             for r in 0:5
                 dP = BSplineDerivativeSpace{r}(P)
+                @test (dP == Q) || (r == 0)
+                Q = BasicBSpline.derivative(Q)
                 for t in ts
                     j = intervalindex(dP,t)
                     B = collect(bsplinebasisall(dP,j,t))