diff --git a/docs/DifferentiableProgramming.md b/docs/DifferentiableProgramming.md
index 91b396adada17..0e1196f138cca 100644
--- a/docs/DifferentiableProgramming.md
+++ b/docs/DifferentiableProgramming.md
@@ -1192,7 +1192,14 @@ extension Optional: Differentiable where Wrapped: Differentiable {
 
     @noDerivative
     public var zeroTangentVectorInitializer: () -> TangentVector {
-        { TangentVector(.zero) }
+        switch self {
+        case nil:
+            return { TangentVector(nil) }
+        case let x?:
+            return { [zeroTanInit = x.zeroTangentVectorInitializer] in
+                TangentVector(zeroTanInit())
+            }
+        }
     }
 }
 ```
diff --git a/stdlib/public/Differentiation/ArrayDifferentiation.swift b/stdlib/public/Differentiation/ArrayDifferentiation.swift
index 177e9a3572d6e..046af6f186095 100644
--- a/stdlib/public/Differentiation/ArrayDifferentiation.swift
+++ b/stdlib/public/Differentiation/ArrayDifferentiation.swift
@@ -82,6 +82,12 @@ where Element: Differentiable {
       base[i].move(along: direction.base[i])
     }
   }
+
+  /// A closure that produces a `TangentVector` of zeros with the same
+  /// `count` as `self`.
+  public var zeroTangentVectorInitializer: () -> TangentVector {
+    return base.zeroTangentVectorInitializer
+  }
 }
 
 extension Array.DifferentiableView: Equatable
diff --git a/stdlib/public/Differentiation/CMakeLists.txt b/stdlib/public/Differentiation/CMakeLists.txt
index 0fa1ed82d6c8e..6e0257aa3f73e 100644
--- a/stdlib/public/Differentiation/CMakeLists.txt
+++ b/stdlib/public/Differentiation/CMakeLists.txt
@@ -16,6 +16,7 @@ add_swift_target_library(swift_Differentiation ${SWIFT_STDLIB_LIBRARY_BUILD_TYPE
   DifferentiationUtilities.swift
   AnyDifferentiable.swift
   ArrayDifferentiation.swift
+  OptionalDifferentiation.swift
 
   GYB_SOURCES
     FloatingPointDifferentiation.swift.gyb
diff --git a/stdlib/public/Differentiation/OptionalDifferentiation.swift b/stdlib/public/Differentiation/OptionalDifferentiation.swift
new file mode 100644
index 0000000000000..c8aa82f01e980
--- /dev/null
+++ b/stdlib/public/Differentiation/OptionalDifferentiation.swift
@@ -0,0 +1,83 @@
+//===--- OptionalDifferentiation.swift ------------------------*- swift -*-===//
+//
+// This source file is part of the Swift.org open source project
+//
+// Copyright (c) 2020 Apple Inc. and the Swift project authors
+// Licensed under Apache License v2.0 with Runtime Library Exception
+//
+// See https://swift.org/LICENSE.txt for license information
+// See https://swift.org/CONTRIBUTORS.txt for the list of Swift project authors
+//
+//===----------------------------------------------------------------------===//
+
+import Swift
+
+extension Optional: Differentiable where Wrapped: Differentiable {
+  public struct TangentVector: Differentiable, AdditiveArithmetic {
+    public typealias TangentVector = Self
+
+    public var value: Wrapped.TangentVector?
+
+    public init(_ value: Wrapped.TangentVector?) {
+      self.value = value
+    }
+
+    public static var zero: Self {
+      return Self(.zero)
+    }
+
+    public static func + (lhs: Self, rhs: Self) -> Self {
+      switch (lhs.value, rhs.value) {
+      case (nil, nil): return Self(nil)
+      case let (x?, nil): return Self(x)
+      case let (nil, y?): return Self(y)
+      case let (x?, y?): return Self(x + y)
+      }
+    }
+
+    public static func - (lhs: Self, rhs: Self) -> Self {
+      switch (lhs.value, rhs.value) {
+      case (nil, nil): return Self(nil)
+      case let (x?, nil): return Self(x)
+      case let (nil, y?): return Self(.zero - y)
+      case let (x?, y?): return Self(x - y)
+      }
+    }
+
+    public mutating func move(along direction: TangentVector) {
+      if let value = direction.value {
+        self.value?.move(along: value)
+      }
+    }
+
+    @noDerivative
+    public var zeroTangentVectorInitializer: () -> TangentVector {
+      switch value {
+      case nil:
+        return { Self(nil) }
+      case let x?:
+        return { [zeroTanInit = x.zeroTangentVectorInitializer] in
+          Self(zeroTanInit())
+        }
+      }
+    }
+  }
+
+  public mutating func move(along direction: TangentVector) {
+    if let value = direction.value {
+      self?.move(along: value)
+    }
+  }
+
+  @noDerivative
+  public var zeroTangentVectorInitializer: () -> TangentVector {
+    switch self {
+    case nil:
+      return { TangentVector(nil) }
+    case let x?:
+      return { [zeroTanInit = x.zeroTangentVectorInitializer] in
+        TangentVector(zeroTanInit())
+      }
+    }
+  }
+}
diff --git a/test/AutoDiff/stdlib/optional.swift b/test/AutoDiff/stdlib/optional.swift
new file mode 100644
index 0000000000000..5039f8098294e
--- /dev/null
+++ b/test/AutoDiff/stdlib/optional.swift
@@ -0,0 +1,103 @@
+// RUN: %target-run-simple-swift
+// REQUIRES: executable_test
+
+import _Differentiation
+import StdlibUnittest
+
+var OptionalDifferentiationTests = TestSuite("OptionalDifferentiation")
+
+OptionalDifferentiationTests.test("Optional operations") {
+  // Differentiable.move(along:)
+  do {
+    var some: Float? = 2
+    some.move(along: .init(3))
+    expectEqual(5, some)
+
+    var none: Float? = nil
+    none.move(along: .init(3))
+    expectEqual(nil, none)
+  }
+
+  // Differentiable.zeroTangentVectorInitializer
+  do {
+    let some: [Float]? = [1, 2, 3]
+    expectEqual(.init([0, 0, 0]), some.zeroTangentVectorInitializer())
+
+    let none: [Float]? = nil
+    expectEqual(.init(nil), none.zeroTangentVectorInitializer())
+  }
+}
+
+OptionalDifferentiationTests.test("Optional.TangentVector operations") {
+  // Differentiable.move(along:)
+  do {
+    var some: Optional<Float>.TangentVector = .init(2)
+    some.move(along: .init(3))
+    expectEqual(5, some.value)
+
+    var none: Optional<Float>.TangentVector = .init(nil)
+    none.move(along: .init(3))
+    expectEqual(nil, none.value)
+
+    var nestedSome: Optional<Optional<Float>>.TangentVector = .init(.init(2))
+    nestedSome.move(along: .init(.init(3)))
+    expectEqual(.init(5), nestedSome.value)
+
+    var nestedNone: Optional<Optional<Float>>.TangentVector = .init(.init(nil))
+    nestedNone.move(along: .init(.init(3)))
+    expectEqual(.init(nil), nestedNone.value)
+  }
+
+  // Differentiable.zeroTangentVectorInitializer
+  do {
+    let some: [Float]? = [1, 2, 3]
+    expectEqual(.init([0, 0, 0]), some.zeroTangentVectorInitializer())
+
+    let none: [Float]? = nil
+    expectEqual(.init(nil), none.zeroTangentVectorInitializer())
+
+    let nestedSome: [Float]?? = [1, 2, 3]
+    expectEqual(.init(.init([0, 0, 0])), nestedSome.zeroTangentVectorInitializer())
+
+    let nestedNone: [Float]?? = nil
+    expectEqual(.init(nil), nestedNone.zeroTangentVectorInitializer())
+  }
+
+  // AdditiveArithmetic.zero
+  expectEqual(.init(Float.zero), Float?.TangentVector.zero)
+  expectEqual(.init([Float].TangentVector.zero), [Float]?.TangentVector.zero)
+
+  expectEqual(.init(.init(Float.zero)), Float??.TangentVector.zero)
+  expectEqual(.init(.init([Float].TangentVector.zero)), [Float]??.TangentVector.zero)
+
+  // AdditiveArithmetic.+, AdditiveArithmetic.-
+  do {
+    let some: Optional<Float>.TangentVector = .init(2)
+    let none: Optional<Float>.TangentVector = .init(nil)
+
+    expectEqual(.init(4), some + some)
+    expectEqual(.init(2), some + none)
+    expectEqual(.init(2), none + some)
+    expectEqual(.init(nil), none + none)
+
+    expectEqual(.init(0), some - some)
+    expectEqual(.init(2), some - none)
+    expectEqual(.init(-2), none - some)
+    expectEqual(.init(nil), none - none)
+
+    let nestedSome: Optional<Optional<Float>>.TangentVector = .init(.init(2))
+    let nestedNone: Optional<Optional<Float>>.TangentVector = .init(.init(nil))
+
+    expectEqual(.init(.init(4)), nestedSome + nestedSome)
+    expectEqual(.init(.init(2)), nestedSome + nestedNone)
+    expectEqual(.init(.init(2)), nestedNone + nestedSome)
+    expectEqual(.init(.init(nil)), nestedNone + nestedNone)
+
+    expectEqual(.init(.init(0)), nestedSome - nestedSome)
+    expectEqual(.init(.init(2)), nestedSome - nestedNone)
+    expectEqual(.init(.init(-2)), nestedNone - nestedSome)
+    expectEqual(.init(.init(nil)), nestedNone - nestedNone)
+  }
+}
+
+runAllTests()