diff --git a/stdlib/public/Differentiation/ArrayDifferentiation.swift b/stdlib/public/Differentiation/ArrayDifferentiation.swift
index a4b1149a20f56..177e9a3572d6e 100644
--- a/stdlib/public/Differentiation/ArrayDifferentiation.swift
+++ b/stdlib/public/Differentiation/ArrayDifferentiation.swift
@@ -42,6 +42,14 @@ where Element: Differentiable {
     return (base, { $0 })
   }
 
+  @usableFromInline
+  @derivative(of: base)
+  func _jvpBase() -> (
+    value: [Element], differential: (Array<Element>.TangentVector) -> TangentVector
+  ) {
+    return (base, { $0 })
+  }
+
   /// Creates a differentiable view of the given array.
   public init(_ base: [Element]) { self._base = base }
 
@@ -53,6 +61,14 @@ where Element: Differentiable {
     return (Array.DifferentiableView(base), { $0 })
   }
 
+  @usableFromInline
+  @derivative(of: init(_:))
+  static func _jvpInit(_ base: [Element]) -> (
+    value: Array.DifferentiableView, differential: (TangentVector) -> TangentVector
+  ) {
+    return (Array.DifferentiableView(base), { $0 })
+  }
+
   public typealias TangentVector =
     Array<Element.TangentVector>.DifferentiableView
 
@@ -191,6 +207,17 @@ extension Array where Element: Differentiable {
     return (self[index], pullback)
   }
 
+  @usableFromInline
+  @derivative(of: subscript)
+  func _jvpSubscript(index: Int) -> (
+    value: Element, differential: (TangentVector) -> Element.TangentVector
+  ) {
+    func differential(_ v: TangentVector) -> Element.TangentVector {
+      return v[index]
+    }
+    return (self[index], differential)
+  }
+
   @usableFromInline
   @derivative(of: +)
   static func _vjpConcatenate(_ lhs: Self, _ rhs: Self) -> (
@@ -210,8 +237,26 @@ extension Array where Element: Differentiable {
     }
     return (lhs + rhs, pullback)
   }
+
+  @usableFromInline
+  @derivative(of: +)
+  static func _jvpConcatenate(_ lhs: Self, _ rhs: Self) -> (
+    value: Self,
+    differential: (TangentVector, TangentVector) -> TangentVector
+  ) {
+    func differential(_ l: TangentVector, _ r: TangentVector) -> TangentVector {
+      precondition(
+        l.base.count == lhs.count && r.base.count == rhs.count, """
+          Tangent vectors with invalid count; expected to equal the \
+          operand counts \(lhs.count) and \(rhs.count)
+          """)
+      return .init(l.base + r.base)
+    }
+    return (lhs + rhs, differential)
+  }
 }
 
+
 extension Array where Element: Differentiable {
   @usableFromInline
   @derivative(of: append)
@@ -277,6 +322,17 @@ extension Array where Element: Differentiable {
       }
     )
   }
+
+  @usableFromInline
+  @derivative(of: init(repeating:count:))
+  static func _jvpInit(repeating repeatedValue: Element, count: Int) -> (
+    value: Self, differential: (Element.TangentVector) -> TangentVector
+  ) {
+    (
+      value: Self(repeating: repeatedValue, count: count),
+      differential: { v in TangentVector(.init(repeating: v, count: count)) }
+    )
+  }
 }
 
 //===----------------------------------------------------------------------===//
@@ -312,6 +368,27 @@ extension Array where Element: Differentiable {
     }
     return (value: values, pullback: pullback)
   }
+
+  @inlinable
+  @derivative(of: differentiableMap)
+  internal func _jvpDifferentiableMap<Result: Differentiable>(
+    _ body: @differentiable (Element) -> Result
+  ) -> (
+    value: [Result],
+    differential: (Array.TangentVector) -> Array<Result>.TangentVector
+  ) {
+    var values: [Result] = []
+    var differentials: [(Element.TangentVector) -> Result.TangentVector] = []
+    for x in self {
+      let (y, df) = valueWithDifferential(at: x, in: body)
+      values.append(y)
+      differentials.append(df)
+    }
+    func differential(_ tans: Array.TangentVector) -> Array<Result>.TangentVector {
+      .init(zip(tans.base, differentials).map { tan, df in df(tan) })
+    }
+    return (value: values, differential: differential)
+  }
 }
 
 extension Array where Element: Differentiable {
@@ -361,4 +438,33 @@ extension Array where Element: Differentiable {
       }
     )
   }
+
+  @inlinable
+  @derivative(of: differentiableReduce, wrt: (self, initialResult))
+  func _jvpDifferentiableReduce<Result: Differentiable>(
+    _ initialResult: Result,
+    _ nextPartialResult: @differentiable (Result, Element) -> Result
+  ) -> (value: Result,
+        differential: (Array.TangentVector, Result.TangentVector)
+          -> Result.TangentVector) {
+    var differentials:
+      [(Result.TangentVector, Element.TangentVector) -> Result.TangentVector]
+        = []
+    let count = self.count
+    differentials.reserveCapacity(count)
+    var result = initialResult
+    for element in self {
+      let (y, df) =
+        valueWithDifferential(at: result, element, in: nextPartialResult)
+      result = y
+      differentials.append(df)
+    }
+    return (value: result, differential: { dSelf, dInitial in
+      var dResult = dInitial
+      for (dElement, df) in zip(dSelf.base, differentials) {
+        dResult = df(dResult, dElement)
+      }
+      return dResult
+    })
+  }
 }
diff --git a/test/AutoDiff/validation-test/forward_mode.swift b/test/AutoDiff/validation-test/forward_mode.swift
index 22b049158c432..e43740c1244a5 100644
--- a/test/AutoDiff/validation-test/forward_mode.swift
+++ b/test/AutoDiff/validation-test/forward_mode.swift
@@ -1319,4 +1319,112 @@ ForwardModeTests.test("ForceUnwrapping") {
   expectEqual(5, forceUnwrap(Float(2)))
 }
 
+//===----------------------------------------------------------------------===//
+// Array methods from ArrayDifferentiation.swift
+//===----------------------------------------------------------------------===//
+
+typealias FloatArrayTan = Array<Float>.TangentVector
+
+ForwardModeTests.test("Array.+") {
+  func sumFirstThreeConcatenating(_ a: [Float], _ b: [Float]) -> Float {
+    let c = a + b
+    return c[0] + c[1] + c[2]
+  }
+
+  expectEqual(3, differential(at: [0, 0], [0, 0], in: sumFirstThreeConcatenating)(.init([1, 1]), .init([1, 1])))
+  expectEqual(0, differential(at: [0, 0], [0, 0], in: sumFirstThreeConcatenating)(.init([0, 0]), .init([0, 1])))
+  expectEqual(1, differential(at: [0, 0], [0, 0], in: sumFirstThreeConcatenating)(.init([0, 1]), .init([0, 1])))
+  expectEqual(1, differential(at: [0, 0], [0, 0], in: sumFirstThreeConcatenating)(.init([1, 0]), .init([0, 1])))
+  expectEqual(1, differential(at: [0, 0], [0, 0], in: sumFirstThreeConcatenating)(.init([0, 0]), .init([1, 1])))
+  expectEqual(2, differential(at: [0, 0], [0, 0], in: sumFirstThreeConcatenating)(.init([1, 1]), .init([0, 1])))
+
+  expectEqual(
+    3,
+    differential(at: [0, 0, 0, 0], [0, 0], in: sumFirstThreeConcatenating)(.init([1, 1, 1, 1]), .init([1, 1])))
+  expectEqual(
+    3,
+    differential(at: [0, 0, 0, 0], [0, 0], in: sumFirstThreeConcatenating)(.init([1, 1, 1, 0]), .init([0, 0])))
+  
+  expectEqual(
+    3,
+    differential(at: [], [0, 0, 0, 0], in: sumFirstThreeConcatenating)(.init([]), .init([1, 1, 1, 1])))
+  expectEqual(
+    0,
+    differential(at: [], [0, 0, 0, 0], in: sumFirstThreeConcatenating)(.init([]), .init([0, 0, 0, 1])))
+}
+
+ForwardModeTests.test("Array.init(repeating:count:)") {
+  @differentiable
+  func repeating(_ x: Float) -> [Float] {
+    Array(repeating: x, count: 10)
+  }
+  expectEqual(Float(10), derivative(at: .zero) { x in
+    repeating(x).differentiableReduce(0, {$0 + $1})
+  })
+  expectEqual(Float(20), differential(at: .zero, in: { x in
+    repeating(x).differentiableReduce(0, {$0 + $1})
+  })(2))
+}
+
+ForwardModeTests.test("Array.DifferentiableView.init") {
+  @differentiable
+  func constructView(_ x: [Float]) -> Array<Float>.DifferentiableView {
+    return Array<Float>.DifferentiableView(x)
+  }
+
+  let forward = differential(at: [5, 6, 7, 8], in: constructView)
+  expectEqual(
+    FloatArrayTan([1, 2, 3, 4]),
+    forward(FloatArrayTan([1, 2, 3, 4])))
+}
+
+ForwardModeTests.test("Array.DifferentiableView.base") {
+  @differentiable
+  func accessBase(_ x: Array<Float>.DifferentiableView) -> [Float] {
+    return x.base
+  }
+
+  let forward = differential(
+    at: Array<Float>.DifferentiableView([5, 6, 7, 8]),
+    in: accessBase)
+  expectEqual(
+    FloatArrayTan([1, 2, 3, 4]),
+    forward(FloatArrayTan([1, 2, 3, 4])))
+}
+
+ForwardModeTests.test("Array.differentiableMap") {
+  let x: [Float] = [1, 2, 3]
+  let tan = Array<Float>.TangentVector([1, 1, 1])
+
+  func multiplyMap(_ a: [Float]) -> [Float] {
+    return a.differentiableMap({ x in 3 * x })
+  }
+  expectEqual([3, 3, 3], differential(at: x, in: multiplyMap)(tan))
+
+  func squareMap(_ a: [Float]) -> [Float] {
+    return a.differentiableMap({ x in x * x })
+  }
+  expectEqual([2, 4, 6], differential(at: x, in: squareMap)(tan))
+}
+
+ForwardModeTests.test("Array.differentiableReduce") {
+  let x: [Float] = [1, 2, 3]
+  let tan = Array<Float>.TangentVector([1, 1, 1])
+
+  func sumReduce(_ a: [Float]) -> Float {
+    return a.differentiableReduce(0, { $0 + $1 })
+  }
+  expectEqual(1 + 1 + 1, differential(at: x, in: sumReduce)(tan))
+
+  func productReduce(_ a: [Float]) -> Float {
+    return a.differentiableReduce(1, { $0 * $1 })
+  }
+  expectEqual(x[1] * x[2] + x[0] * x[2] + x[0] * x[1], differential(at: x, in: productReduce)(tan))
+
+  func sumOfSquaresReduce(_ a: [Float]) -> Float {
+    return a.differentiableReduce(0, { $0 + $1 * $1 })
+  }
+  expectEqual(2 * x[0] + 2 * x[1] + 2 * x[2], differential(at: x, in: sumOfSquaresReduce)(tan))
+}
+
 runAllTests()