diff --git a/include/backwardad.hpp b/include/backwardad.hpp
new file mode 100644
index 0000000..edc85e4
--- /dev/null
+++ b/include/backwardad.hpp
@@ -0,0 +1,56 @@
+#pragma once
+#include <vector>
+#include <cmath>
+#include "dual.hpp"
+
+namespace backwardad {
+
+inline void topo_sort(const Node& n, std::vector<Node>& order, std::vector<Node>& vis) {
+    auto same = [&](const Node& x) { return x.ptr.get() == n.ptr.get(); };
+    if (std::find_if(vis.begin(), vis.end(), same) != vis.end()) return;
+    vis.push_back(n);
+    for (const auto& in : n.inputs()) topo_sort(in, order, vis);
+    order.push_back(n);
+}
+
+inline void backward(const Node& output) {
+// 第一步：拓扑排序（保证从后往前算）
+    std::vector<Node> order;
+    std::vector<Node> visited;
+    topo_sort(output, order, visited);
+
+    // 第二步：初始化输出梯度 = 1
+    output.set_gradient(1.0);
+    std::reverse(order.begin(), order.end());
+
+    // 第三步：反向遍历，逐个调用节点的局部导数
+    for (const auto& v : order) {
+        if (v.backward()) v.backward()();
+    }
+}
+
+template <typename Func, typename... Args>
+ADResult diff(const Func f, Args... args) {
+    ADResult res;
+
+    // 1. 创建输入节点
+    std::vector<Node> inputs;
+    inputs.reserve(sizeof...(Args));
+    (inputs.emplace_back((double)args), ...);
+
+    // 2. 前向计算
+    Node output = [&]<size_t... Is>(std::index_sequence<Is...>) {
+        return f(inputs[Is]...);
+    }(std::make_index_sequence<sizeof...(Args)>{});
+
+    // 3. 反向传播
+    backward(output);
+
+    // 4. 提取结果
+    res.value = output.value();
+    for (auto& in : inputs)
+        res.gradient.push_back(in.gradient());
+
+    return res;
+}
+} // namespace backwardad
\ No newline at end of file
diff --git a/include/dual.hpp b/include/dual.hpp
index eb2ea6e..b112a8d 100644
--- a/include/dual.hpp
+++ b/include/dual.hpp
@@ -3,6 +3,7 @@
 #include <iostream>
 #include <vector>
 #include <cmath>
+#include "types/common.hpp"
 #include "types/dual.hpp"
 
 // 前向自动微分的运算部分，两个deriv相乘为0
@@ -62,4 +63,123 @@ namespace forwardad{
     inline Dual atan(const Dual& x) {
         return Dual(std::atan(x.value), x.deriv / (1 + x.value * x.value));
     }
-}// namespace forwardad
\ No newline at end of file
+}// namespace forwardad
+
+namespace backwardad {
+    inline Node operator+(const Node& a, const Node& b) {
+        Node out(a.value() + b.value());
+        out.inputs() = {a, b};
+        out.backward() = [out, a, b]() {
+            a.add_gradient(out.gradient()); // da = dout * 1
+            b.add_gradient(out.gradient()); // db = dout * 1
+        };
+        return out;
+    }
+
+    inline Node operator-(const Node& a, const Node& b) {
+        Node out(a.value() - b.value());
+        out.inputs() = {a, b};
+        out.backward() = [out, a, b]() {
+            a.add_gradient(out.gradient()); // da = dout * 1
+            b.add_gradient(-out.gradient()); // db = dout * (-1)
+        };
+        return out;
+    }
+
+    inline Node operator*(const Node& a, const Node& b) {
+        Node out(a.value() * b.value());
+        out.inputs() = {a, b};
+        out.backward() = [out, a, b]() {
+            a.add_gradient(out.gradient() * b.value()); // da = dout * b
+            b.add_gradient(out.gradient() * a.value()); // db = dout * a
+        };
+        return out;
+    }
+
+    inline Node operator/(const Node& a, const Node& b) {
+        Node out(a.value() / b.value());
+        out.inputs() = {a, b};
+        out.backward() = [out, a, b]() {
+            a.add_gradient(out.gradient() / b.value()); // da = dout * (1/b)
+            b.add_gradient(-out.gradient() * a.value() / (b.value() * b.value())); // db = dout * (-a/b^2)
+        };
+        return out;
+    }
+
+    // 下面的函数需要用到链式法则(使用泰勒展开后保留一次项)
+    // 三角函数
+    inline Node sin(const Node& x) {
+        Node out(std::sin(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(out.gradient() * std::cos(x.value())); // dx = dout * cos(x)
+        };
+        return out;
+    }
+
+    inline Node cos(const Node& x) {
+        Node out(std::cos(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(-out.gradient() * std::sin(x.value())); // dx = dout * (-sin(x))
+        };
+        return out;
+    }
+
+    // 指数和对数
+    inline Node exp(const Node& x) {
+        Node out(std::exp(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(out.gradient() * out.value()); // dx = dout * exp(x) = dout * out.value
+        };
+        return out;
+    }
+
+    inline Node log(const Node& x) {
+        Node out(std::log(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(out.gradient() / x.value()); // dx = dout * (1/x)
+        };
+        return out;
+    }
+
+    inline Node pow(const Node& x, double n) {
+        Node out(std::pow(x.value(), n));
+        out.inputs() = {x};
+        out.backward() = [out, x, n]() {
+            x.add_gradient(out.gradient() * n * std::pow(x.value(), n - 1)); // dx = dout * (n * x^(n-1))
+        };
+        return out;
+    }
+
+    // 反三角函数
+    inline Node asin(const Node& x) {
+        Node out(std::asin(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(out.gradient() / std::sqrt(1 - x.value() * x.value())); // dx = dout * (1/sqrt(1-x^2))
+        };
+        return out;
+    }
+
+    inline Node acos(const Node& x) {
+        Node out(std::acos(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(-out.gradient() / std::sqrt(1 - x.value() * x.value())); // dx = dout * (-1/sqrt(1-x^2))
+        };
+        return out;
+    }
+
+    inline Node atan(const Node& x) {
+        Node out(std::atan(x.value()));
+        out.inputs() = {x};
+        out.backward() = [out, x]() {
+            x.add_gradient(out.gradient() / (1 + x.value() * x.value())); // dx = dout * (1/(1+x^2))
+        };
+        return out;
+    }
+
+}
\ No newline at end of file
diff --git a/include/forwardad.hpp b/include/forwardad.hpp
index a1385de..c1b125a 100644
--- a/include/forwardad.hpp
+++ b/include/forwardad.hpp
@@ -9,8 +9,8 @@
 namespace forwardad{
 
 template <typename Func, typename... Args>
-Result diff(const Func& f, Args... args) {
-    Result res;
+ADResult diff(const Func& f, Args... args) {
+    ADResult res;
     constexpr size_t N = sizeof...(Args);
     res.gradient.resize(N);
 
diff --git a/include/types/common.hpp b/include/types/common.hpp
index 41ff214..a084212 100644
--- a/include/types/common.hpp
+++ b/include/types/common.hpp
@@ -1,9 +1,9 @@
 /*返回数据结构体*/
+#pragma once
+#include <functional>
 #include <vector>
 
-namespace forwardad{
-    struct Result{
-        double value; // 函数值
-        std::vector<double> gradient; // 梯度
-    };
-}
\ No newline at end of file
+struct ADResult{
+    double value; // 函数值
+    std::vector<double> gradient; // 梯度
+};
diff --git a/include/types/dual.hpp b/include/types/dual.hpp
index 630588c..f8294cf 100644
--- a/include/types/dual.hpp
+++ b/include/types/dual.hpp
@@ -1,5 +1,9 @@
 /*对偶数的数据类型定义部分*/
 #pragma once
+#include <iostream>
+#include <functional>
+#include <memory>
+#include <vector>
 
 namespace forwardad {
 struct Dual {
@@ -9,4 +13,43 @@ struct Dual {
     Dual(double v = 0.0, double d = 0.0) : value(v), deriv(d) {}
 
 };
-} // namespace forwardad
\ No newline at end of file
+} // namespace forwardad
+
+namespace backwardad{
+    struct NodeData;
+
+    struct Node{
+        std::shared_ptr<NodeData> ptr;
+
+        Node() = default;
+        explicit Node(double v);
+
+        double value() const;
+        double gradient() const;
+        void add_gradient(double g) const;
+        void set_gradient(double g) const;
+        const std::vector<Node>& inputs() const;
+        std::vector<Node>& inputs();
+        std::function<void()>& backward();
+        const std::function<void()>& backward() const;
+    };
+
+    struct NodeData {
+        double value;
+        double gradient;
+        std::vector<Node> inputs;
+        std::function<void()> backward;
+
+        explicit NodeData(double v) : value(v), gradient(0.0) {}
+    };
+
+    inline Node::Node(double v) : ptr(std::make_shared<NodeData>(v)) {}
+    inline double Node::value() const { return ptr->value; }
+    inline double Node::gradient() const { return ptr->gradient; }
+    inline void Node::add_gradient(double g) const { ptr->gradient += g; }
+    inline void Node::set_gradient(double g) const { ptr->gradient = g; }
+    inline const std::vector<Node>& Node::inputs() const { return ptr->inputs; }
+    inline std::vector<Node>& Node::inputs() { return ptr->inputs; }
+    inline std::function<void()>& Node::backward() { return ptr->backward; }
+    inline const std::function<void()>& Node::backward() const { return ptr->backward; }
+} // namespace backwardad
\ No newline at end of file
diff --git a/tests/basic.cpp b/tests/basic.cpp
index c734a15..f669492 100644
--- a/tests/basic.cpp
+++ b/tests/basic.cpp
@@ -1,40 +1,75 @@
 #include <iostream>
 #include "forwardad.hpp"
+#include "backwardad.hpp"
 #include "types/dual.hpp"
 
-using namespace forwardad;
-
 // 一阶测试函数
-Dual f(Dual x) {
+forwardad::Dual f1(forwardad::Dual x) {
     return x + x*x + pow(x,3);
 }
 
 // 二阶测试函数
-Dual g(Dual x, Dual y) {
+forwardad::Dual g1(forwardad::Dual x, forwardad::Dual y) {
     return exp(x) * log(y);
 }
 
 // 高阶测试函数
-Dual h(Dual x, Dual y, Dual z) {
+forwardad::Dual h1(forwardad::Dual x, forwardad::Dual y, forwardad::Dual z) {
     return pow(x, 3) + pow(y, 2) + z + cos(x * y * z);
 }
 
-int main() {
+backwardad::Node f2(backwardad::Node x) {
+    return x + x*x + pow(x,3);
+}
+
+backwardad::Node g2(backwardad::Node x, backwardad::Node y) {
+    return exp(x) * log(y);
+}   
+
+backwardad::Node h2(backwardad::Node x, backwardad::Node y, backwardad::Node z) {
+    return pow(x, 3) + pow(y, 2) + z + cos(x * y * z);
+}
+
+int test_forward_ad() {
     std::cout << "Testing f(x) = x^2 + sin(x) at x=2.0\n";
-    auto r = diff(f, 2.0);
+    auto r = forwardad::diff(f1, 2.0);
     std::cout << "value = " << r.value << "\n";
     std::cout << "grad  = " << r.gradient[0] << "\n";
 
     std::cout << "\nTesting g(x,y) = exp(x)*log(y) at (x,y)=(1.0, 2.0)\n";
-    auto r2 = diff(g, 1.0, 2.0);
+    auto r2 = forwardad::diff(g1, 1.0, 2.0);
     std::cout << "value = " << r2.value << "\n";
     std::cout << "grad  = (" << r2.gradient[0] << ", " << r2.gradient[1] << ")\n";
 
     std::cout << "\nTesting h(x,y,z) = x^3 + y^2 + z + cos(x*y*z) at (x,y,z)=(1.0, 2.0, 3.0)\n";
-    auto r3 = diff(h, 1.0, 2.0, 3.0);
+    auto r3 = forwardad::diff(h1, 1.0, 2.0, 3.0);
     std::cout << "value = " << r3.value << "\n";
     std::cout << "grad  = (" << r3.gradient[0] << ", " << r3.gradient[1] << ", " << r3.gradient[2] << ")\n";
+    return 0;
+}
 
+int test_backward_ad() {
+    std::cout << "Testing f(x) = x^2 + sin(x) at x=2.0\n";
+    auto r = backwardad::diff(f2, 2.0);
+    std::cout << "value = " << r.value << "\n";
+    std::cout << "grad  = " << r.gradient[0] << "\n";
 
+    std::cout << "\nTesting g(x,y) = exp(x)*log(y) at (x,y)=(1.0, 2.0)\n";
+    auto r2 = backwardad::diff(g2, 1.0, 2.0);
+    std::cout << "value = " << r2.value << "\n";
+    std::cout << "grad  = (" << r2.gradient[0] << ", " << r2.gradient[1] << ")\n";
+
+    std::cout << "\nTesting h(x,y,z) = x^3 + y^2 + z + cos(x*y*z) at (x,y,z)=(1.0, 2.0, 3.0)\n";
+    auto r3 = backwardad::diff(h2, 1.0, 2.0, 3.0);
+    std::cout << "value = " << r3.value << "\n";
+    std::cout << "grad  = (" << r3.gradient[0] << ", " << r3.gradient[1] << ", " << r3.gradient[2] << ")\n";
+    return 0;
+}
+
+int main() {
+    std::cout << "=== Forward AD Tests ===\n";
+    test_forward_ad();
+    std::cout << "\n=== Backward AD Tests ===\n";
+    test_backward_ad();
     return 0;
 }
\ No newline at end of file