Partial derivative of a polynomial using Doubly Linked List

Last Updated : 23 Jul, 2025

Given a 2-variable polynomial represented by a doubly linked list, the task is to find the partial derivative of a polynomial stored in the doubly-linked list.

Examples:

Input: P(x, y) = 2(x^3 y^4) + 3(x^5 y^7) + 1(x^2 y^6)
Output:
Partial derivatives w.r.t. x: 6(x^2 y^4) + 15(x^4 y^7) + 2(x^1 y^6)
Partial derivatives w.r.t. y: 24(x^2 y^3) + 105(x^4 y^6) + 12(x^1 y^5)
Partial derivatives w.r.t. x and y: 144(x^1 y^2) + 2520(x^3 y^5) + 60(x^0 y^4)

Input: P(x, y) = 3(x^2 y^1) + 4(x^2 y^3) + 2(x^4 y^7)
Output:
Partial derivatives w.r.t. x: 6(x^1 y^1) + 8(x^1 y^3) + 8(x^3 y^7)
Partial derivatives w.r.t. y: 6(x^1 y^0) + 24(x^1 y^2) + 56(x^3 y^6)
Partial derivatives w.r.t. x and y: 48(x^0 y^1) + 1008(x^2 y^5)

Approach: Follow the steps below to solve this problem:

Declare a class or structure to store the contents of a node, i.e. data representing the coefficient, power1 representing the power to which x is raised, power2 representing the power to which y is raised, and the pointers to its next and previous node.
Declare functions to calculate derivatives with respect to x, derivative with respect to y, and derivative with respect to x and y.
Calculate and print the derivatives obtained.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Structure of a node
struct node {
    node* link1 = NULL;
    node* link2 = NULL;
    int data = 0;
    int pow1 = 0;
    int pow2 = 0;
};

// Function to generate Doubly Linked
// List from given parameters
void input_equation(node*& head, int d,
                    int p1, int p2)
{
    node* temp = head;

    // If list is empty
    if (head == NULL) {

        // Create new node
        node* ptr = new node();
        ptr->data = d;
        ptr->pow1 = p1;
        ptr->pow2 = p2;

        // Set it as the head
        // of the linked list
        head = ptr;
    }

    // If list is not empty
    else {

        // Temporarily store
        // address of the head node
        temp = head;

        // Traverse the linked list
        while (temp->link2 != NULL) {

            // Link to next node
            temp = temp->link2;
        }

        // Create new node
        node* ptr = new node();
        ptr->data = d;
        ptr->pow1 = p1;
        ptr->pow2 = p2;

        // Connect the nodes
        ptr->link1 = temp;
        temp->link2 = ptr;
    }
}

// Function to calculate partial
// derivative w.r.t. X
void derivation_with_x(node*& head)
{
    cout << "Partial derivatives"
         << " w.r.t. x: ";

    node* temp = head;

    // Traverse the Linked List
    while (temp != NULL) {

        if (temp->pow1 != 0) {
            temp->data = (temp->data)
                         * (temp->pow1);
            temp->pow1 = temp->pow1 - 1;
        }
        else {
            temp->data = 0;
            temp->pow1 = 0;
            temp->pow2 = 0;
        }

        temp = temp->link2;
    }

    temp = head;

    cout << " " << temp->data
         << "(x^" << temp->pow1
         << " y^" << temp->pow2
         << ")";
    temp = temp->link2;

    while (temp != NULL) {
        cout << " + "
             << temp->data << "(x^"
             << temp->pow1 << " y^"
             << temp->pow2 << ")";
        temp = temp->link2;
    }

    cout << "\n";
}

// Function to calculate partial
// derivative w.r.t. Y
void derivation_with_y(node*& head)
{
    cout << "Partial derivatives"
         << " w.r.t. y: ";

    node* temp = head;

    // Traverse the Linked List
    while (temp != NULL) {

        if (temp->pow2 != 0) {
            temp->data = (temp->data)
                         * (temp->pow2);
            temp->pow2 = temp->pow2 - 1;
        }
        else {
            temp->data = 0;
            temp->pow1 = 0;
            temp->pow2 = 0;
        }

        temp = temp->link2;
    }

    temp = head;
    cout << " "
         << temp->data
         << "(x^" << temp->pow1
         << " y^"
         << temp->pow2 << ")";
    temp = temp->link2;

    while (temp != NULL) {
        cout << " + "
             << temp->data << "(x^"
             << temp->pow1 << " y^"
             << temp->pow2 << ")";
        temp = temp->link2;
    }
    cout << "\n";
}

// Function to calculate partial
// derivative w.r.t. XY
void derivation_with_x_y(node*& head)
{
    cout << "Partial derivatives"
         << " w.r.t. x and y: ";

    node* temp = head;

    // Derivative with respect to
    // the first variable
    while (temp != NULL) {
        if (temp->pow1 != 0) {

            temp->data = (temp->data)
                         * (temp->pow1);
            temp->pow1 = temp->pow1 - 1;
        }

        else {
            temp->data = 0;
            temp->pow1 = 0;
            temp->pow2 = 0;
        }

        temp = temp->link2;
    }
    temp = head;

    // Derivative with respect to
    // the second variable
    while (temp != NULL) {

        if (temp->pow2 != 0) {
            temp->data = (temp->data)
                         * (temp->pow2);
            temp->pow2 = temp->pow2 - 1;
        }

        else {
            temp->data = 0;
            temp->pow1 = 0;
            temp->pow2 = 0;
        }

        temp = temp->link2;
    }

    temp = head;
    cout << " "
         << temp->data << "(x^"
         << temp->pow1 << " y^"
         << temp->pow2 << ")";

    temp = temp->link2;

    // Print the list after the
    // calculating the derivative
    while (temp != NULL) {

        cout << " + "
             << temp->data << "(x^"
             << temp->pow1 << " y^"
             << temp->pow2 << ")";
        temp = temp->link2;
    }
    cout << "\n";
}

// Driver Code
int main()
{
    node* head1 = NULL;

    // Creating doubly-linked list
    input_equation(head1, 2, 3, 4);
    input_equation(head1, 3, 5, 7);
    input_equation(head1, 1, 2, 6);

    // Function Call
    derivation_with_x(head1);
    derivation_with_y(head1);
    derivation_with_x_y(head1);

    return 0;
}

Java

import java.util.*;

// Java code
// Structure of a node
class Node {

    public int data;
    public int pow1;
    public int pow2;
    public Node link1;
    public Node link2;

    public Node(int d, int p1, int p2)
    {
        data = d;
        pow1 = p1;
        pow2 = p2;
        link1 = null;
        link2 = null;
    }
}

class GFG {

    // Function to generate Doubly Linked
    // List from given parameters
    public static Node input_equation(Node head, int d,
                                      int p1, int p2)
    {
        Node temp = head;
        // If list is empty
        if (head == null) {

            // Set it as the head
            // of the linked list
            Node ptr = new Node(d, p1, p2);
            head = ptr;
            // If list is not empty
        }
        else {
            // Temporarily store
            // address of the head node
            temp = head;
            while (temp.link2 != null) {
                temp = temp.link2;
            }
            Node ptr = new Node(d, p1, p2);
            // Connect the nodes
            ptr.link1 = temp;
            temp.link2 = ptr;
        }
        return head;
    }

    // Function to calculate partial derivative w.r.t. X
    public static void derivation_with_x(Node head)
    {
        String tem = "Partial derivatives w.r.t. x: ";
        Node temp = head;
        while (temp != null) {
            if (temp.pow1 != 0) {
                temp.data = (temp.data) * (temp.pow1);
                temp.pow1 = temp.pow1 - 1;
            }
            else {
                temp.data = 0;
                temp.pow1 = 0;
                temp.pow2 = 0;
            }
            temp = temp.link2;
        }
        temp = head;
        if (temp.data != 0) {
            tem = tem + " " + temp.data + "(x^" + temp.pow1
                  + " y^" + temp.pow2 + ")";
        }
        temp = temp.link2;
        while (temp != null) {
            if (temp.data != 0) {
                tem = tem + " + " + temp.data + "(x^"
                      + temp.pow1 + " y^" + temp.pow2 + ")";
            }
            temp = temp.link2;
        }
        System.out.println(tem);
    }

    // Function to calculate partial
    // derivative w.r.t. Y
    public static void derivation_with_y(Node head)
    {
        String tem = "Partial derivatives w.r.t. y: ";
        Node temp = head;
        while (temp != null) {
            if (temp.pow2 != 0) {
                temp.data = (temp.data) * (temp.pow2);
                temp.pow2 = temp.pow2 - 1;
            }
            else {
                temp.data = 0;
                temp.pow1 = 0;
                temp.pow2 = 0;
            }
            temp = temp.link2;
        }
        temp = head;
        if (temp.data != 0) {
            tem = tem + " " + temp.data + "(x^" + temp.pow1
                  + " y^" + temp.pow2 + ")";
        }
        temp = temp.link2;
        while (temp != null) {
            if (temp.data != 0) {
                tem = tem + " + " + temp.data + "(x^"
                      + temp.pow1 + " y^" + temp.pow2 + ")";
            }
            temp = temp.link2;
        }
        System.out.println(tem);
    }

    // Function to calculate partial
    // derivative w.r.t. XY
    public static void derivation_with_x_y(Node head)
    {
        String tem = "Partial derivatives w.r.t. xy: ";
        Node temp = head;
        while (temp != null) {
            if (temp.pow1 != 0 && temp.pow2 != 0) {
                temp.data = (temp.data) * (temp.pow1)
                            * (temp.pow2);
                temp.pow1 = temp.pow1 - 1;
                temp.pow2 = temp.pow2 - 1;
            }
            else {
                temp.data = 0;
                temp.pow1 = 0;
                temp.pow2 = 0;
            }
            temp = temp.link2;
        }
        temp = head;
        if (temp.data != 0) {
            tem = tem + " " + temp.data + "(x^" + temp.pow1
                  + " y^" + temp.pow2 + ")";
        }
        temp = temp.link2;
        while (temp != null) {
            if (temp.data != 0) {
                tem = tem + " + " + temp.data + "(x^"
                      + temp.pow1 + " y^" + temp.pow2 + ")";
            }
            temp = temp.link2;
        }
        System.out.println(tem);
    }

    public static void main(String[] args)
    {
        // Creating doubly-linked list
        Node head = null;
        head = input_equation(head, 2, 3, 4);
        head = input_equation(head, 3, 5, 7);
        head = input_equation(head, 1, 2, 6);

        // Function Call
        derivation_with_x(head);
        derivation_with_y(head);
        derivation_with_x_y(head);
    }
}

// The code is contributed by phasing17

Python3

# Python code for the  above approach
# Structure of a node
class Node:
    def __init__(self, data, p1, p2):
        self.data = data
        self.pow1 = p1
        self.pow2 = p2
        self.link1 = None
        self.link2 = None
        
# Function to generate Doubly Linked
# List from given parameters
def input_equation(head, d, p1, p2):
    temp = head
    # If list is empty
    if head is None:
      
      # Set it as the head
       # of the linked list
        ptr = Node(d,p1,p2)
        head = ptr
        #If list is not empty
    else:
      # Temporarily store
      # address of the head node
        temp = head
        while temp.link2 is not None:
            temp = temp.link2
            
       
        ptr = Node(d,p1,p2)
        # Connect the nodes
        ptr.link1 = temp
        temp.link2 = ptr
    return head

# Function to calculate partial derivative w.r.t. X
def derivation_with_x(head):
    print("Partial derivatives w.r.t. x: ", end = "")
    temp = head
    while temp is not None:
        if temp.pow1 != 0:
            temp.data = (temp.data) * (temp.pow1)
            temp.pow1 = temp.pow1 - 1
        else:
            temp.data = 0
            temp.pow1 = 0
            temp.pow2 = 0
        temp = temp.link2
    temp = head
    if temp.data != 0:
        print(" " + str(temp.data) + "(x^" + str(temp.pow1) + " y^" + str(temp.pow2) + ")", end = "")
    temp = temp.link2
    while temp is not None:
        if temp.data != 0:
            print(" + " + str(temp.data) + "(x^" + str(temp.pow1) + " y^" + str(temp.pow2) + ")", end = "")
        temp = temp.link2
    print()
    
# Function to calculate partial
# derivative w.r.t. Y
def derivation_with_y(head):
    print("Partial derivatives w.r.t. y: ", end = "")
    temp = head
    while temp is not None:
        if temp.pow2 != 0:
            temp.data = (temp.data) * (temp.pow2)
            temp.pow2 = temp.pow2 - 1
        else:
            temp.data = 0
            temp.pow1 = 0
            temp.pow2 = 0
        temp = temp.link2
    temp = head
    if temp.data != 0:
        print(" " + str(temp.data) + "(x^" + str(temp.pow1) + " y^" + str(temp.pow2) + ")", end = "")
    temp = temp.link2
    while temp is not None:
        if temp.data != 0:
            print(" + " + str(temp.data) + "(x^" + str(temp.pow1) + " y^" + str(temp.pow2) + ")", end = "")
        temp = temp.link2
    print()
    
#Function to calculate partial
# derivative w.r.t. XY
def derivation_with_x_y(head):
    print("Partial derivatives w.r.t. xy: ", end = "")
    temp = head
    while temp is not None:
        if temp.pow1 != 0 and temp.pow2 != 0:
            temp.data = (temp.data) * (temp.pow1) * (temp.pow2)
            temp.pow1 = temp.pow1 - 1
            temp.pow2 = temp.pow2 - 1
        else:
            temp.data = 0
            temp.pow1 = 0
            temp.pow2 = 0
        temp = temp.link2
    temp = head
    if temp.data != 0:
        print(" " + str(temp.data) + "(x^" + str(temp.pow1) + " y^" + str(temp.pow2) + ")", end = "")
    temp = temp.link2
    while temp is not None:
        if temp.data != 0:
            print(" + " + str(temp.data) + "(x^" + str(temp.pow1) + " y^" + str(temp.pow2) + ")", end = "")
        temp = temp.link2
    print()
    
# Driver Code
if __name__ == "__main__":
  
  #Creating doubly-linked list
    head = None
    head = input_equation(head, 2,3,4)
    head = input_equation(head, 3,5,7)
    head = input_equation(head, 1,2,6)
    
    #Function Call
    derivation_with_x(head)
    derivation_with_y(head)
    derivation_with_x_y(head)

# This code is contributed by lokeshpotta20.

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;

// C# code
// Structure of a node
class Node {

  public int data;
  public int pow1;
  public int pow2;
  public Node link1;
  public Node link2;

  public Node(int d,int p1,int p2) {
    data = d;
    pow1 = p1;
    pow2 = p2;
    link1 = null;
    link2 = null;
  }
}


class HelloWorld {


  // Function to generate Doubly Linked
  // List from given parameters
  public static Node input_equation(Node head,int d,int p1,int p2) {
    Node temp = head;
    // If list is empty
    if (head == null) {

      // Set it as the head
      // of the linked list
      Node ptr = new Node(d,p1,p2);
      head = ptr;
      //If list is not empty
    } else {
      // Temporarily store
      // address of the head node
      temp = head;
      while (temp.link2 != null) {
        temp = temp.link2;
      }
      Node ptr = new Node(d,p1,p2);
      // Connect the nodes
      ptr.link1 = temp;
      temp.link2 = ptr;
    }
    return head;
  }

  // Function to calculate partial derivative w.r.t. X
  public static void derivation_with_x(Node head) {
    string tem = "Partial derivatives w.r.t. x: ";
    Node temp = head;
    while (temp != null) {
      if (temp.pow1 != 0) {
        temp.data = (temp.data) * (temp.pow1);
        temp.pow1 = temp.pow1 - 1;
      } else {
        temp.data = 0;
        temp.pow1 = 0;
        temp.pow2 = 0;
      }
      temp = temp.link2;
    }
    temp = head;
    if (temp.data != 0) {
      tem = tem + " " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
    }
    temp = temp.link2;
    while (temp != null) {
      if (temp.data != 0) {
        tem = tem +" + " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
      }
      temp = temp.link2;
    }
    Console.WriteLine(tem);
  }

  // Function to calculate partial
  // derivative w.r.t. Y
  public static void derivation_with_y(Node head) {
    string tem = "Partial derivatives w.r.t. y: ";
    Node temp = head;
    while (temp != null) {
      if (temp.pow2 != 0) {
        temp.data = (temp.data) * (temp.pow2);
        temp.pow2 = temp.pow2 - 1;
      } else {
        temp.data = 0;
        temp.pow1 = 0;
        temp.pow2 = 0;
      }
      temp = temp.link2;
    }
    temp = head;
    if (temp.data != 0) {
      tem = tem + " " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
    }
    temp = temp.link2;
    while (temp != null) {
      if (temp.data != 0) {
        tem = tem + " + " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
      }
      temp = temp.link2;
    }
    Console.WriteLine(tem);
  }

  // Function to calculate partial
  // derivative w.r.t. XY
  public static void derivation_with_x_y(Node head) {
    string tem = "Partial derivatives w.r.t. xy: ";
    Node temp = head;
    while (temp != null) {
      if (temp.pow1 != 0 && temp.pow2 != 0) {
        temp.data = (temp.data) * (temp.pow1) * (temp.pow2);
        temp.pow1 = temp.pow1 - 1;
        temp.pow2 = temp.pow2 - 1;
      } else {
        temp.data = 0;
        temp.pow1 = 0;
        temp.pow2 = 0;
      }
      temp = temp.link2;
    }
    temp = head;
    if (temp.data != 0) {
      tem = tem + " " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
    }
    temp = temp.link2;
    while (temp != null) {
      if (temp.data != 0) {
        tem = tem + " + " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
      }
      temp = temp.link2;
    }
    Console.WriteLine(tem);
  }

  static void Main() {
    // Creating doubly-linked list
    Node head = null;
    head = input_equation(head, 2,3,4);
    head = input_equation(head, 3,5,7);
    head = input_equation(head, 1,2,6);

    //Function Call
    derivation_with_x(head);
    derivation_with_y(head);
    derivation_with_x_y(head);
  }
}

// The code is contributed by Nidhi goel.

JavaScript

// Javascript code
// Structure of a node
class Node {
    constructor(data, p1, p2) {
        this.data = data;
        this.pow1 = p1;
        this.pow2 = p2;
        this.link1 = null;
        this.link2 = null;
    }
}

// Function to generate Doubly Linked
// List from given parameters
function input_equation(head, d, p1, p2) {
    let temp = head;
    // If list is empty
    if (head == null) {
      
      // Set it as the head
       // of the linked list
        let ptr = new Node(d,p1,p2);
        head = ptr;
        //If list is not empty
    } else {
      // Temporarily store
      // address of the head node
        temp = head;
        while (temp.link2 != null) {
            temp = temp.link2;
        }
        let ptr = new Node(d,p1,p2);
        // Connect the nodes
        ptr.link1 = temp;
        temp.link2 = ptr;
    }
    return head;
}

// Function to calculate partial derivative w.r.t. X
function derivation_with_x(head) {
    tem = "Partial derivatives w.r.t. x: ";
    let temp = head;
    while (temp != null) {
        if (temp.pow1 != 0) {
            temp.data = (temp.data) * (temp.pow1);
            temp.pow1 = temp.pow1 - 1;
        } else {
            temp.data = 0;
            temp.pow1 = 0;
            temp.pow2 = 0;
        }
        temp = temp.link2;
    }
    temp = head;
    if (temp.data != 0) {
        tem = tem + " " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
    }
    temp = temp.link2;
    while (temp != null) {
        if (temp.data != 0) {
        tem = tem +" + " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
        }
        temp = temp.link2;
    }
    console.log(tem);
}
    
// Function to calculate partial
// derivative w.r.t. Y
function derivation_with_y(head) {
    tem = "Partial derivatives w.r.t. y: ";
    let temp = head;
    while (temp != null) {
        if (temp.pow2 != 0) {
            temp.data = (temp.data) * (temp.pow2);
            temp.pow2 = temp.pow2 - 1;
        } else {
            temp.data = 0;
            temp.pow1 = 0;
            temp.pow2 = 0;
        }
        temp = temp.link2;
    }
    temp = head;
    if (temp.data != 0) {
        tem = tem + " " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
    }
    temp = temp.link2;
    while (temp != null) {
        if (temp.data != 0) {
            tem = tem + " + " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
        }
        temp = temp.link2;
    }
    console.log(tem);
}
    
// Function to calculate partial
// derivative w.r.t. XY
function derivation_with_x_y(head) {
    tem = "Partial derivatives w.r.t. xy: ";
    let temp = head;
    while (temp != null) {
        if (temp.pow1 != 0 && temp.pow2 != 0) {
            temp.data = (temp.data) * (temp.pow1) * (temp.pow2);
            temp.pow1 = temp.pow1 - 1;
            temp.pow2 = temp.pow2 - 1;
        } else {
            temp.data = 0;
            temp.pow1 = 0;
            temp.pow2 = 0;
        }
        temp = temp.link2;
    }
    temp = head;
    if (temp.data != 0) {
        tem = tem + " " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
    }
    temp = temp.link2;
    while (temp != null) {
        if (temp.data != 0) {
            tem = tem + " + " + temp.data + "(x^" + temp.pow1 + " y^" + temp.pow2 + ")";
        }
        temp = temp.link2;
    }
    console.log(tem);
}
    
// Driver Code
function main() {
  
  // Creating doubly-linked list
    let head = null;
    head = input_equation(head, 2,3,4);
    head = input_equation(head, 3,5,7);
    head = input_equation(head, 1,2,6);
    
    //Function Call
    derivation_with_x(head);
    derivation_with_y(head);
    derivation_with_x_y(head);
}

main();

Output:

Partial derivatives w.r.t. x:  6(x^2 y^4) + 15(x^4 y^7) + 2(x^1 y^6)
Partial derivatives w.r.t. y:  24(x^2 y^3) + 105(x^4 y^6) + 12(x^1 y^5)
Partial derivatives w.r.t. x and y:  144(x^1 y^2) + 2520(x^3 y^5) + 60(x^0 y^4)

Time Complexity: O(N)
Auxiliary Space: O(1)

Analysis of Algorithms

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Partial derivative of a polynomial using Doubly Linked List

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