Approximate a Derivative at a Point in C

Introduction

In this lab, we will learn how to approximate the derivative of a function at a specific point using the C programming language. We will start by defining a simple quadratic function, then use a small value h to compute the numerical derivative using the difference method. Finally, we will print the approximate derivative. This lab aims to provide a practical understanding of derivative approximation, which is a fundamental concept in calculus and analytical geometry.

Skills Graph

Define the Function f(x)

In this step, we will define a mathematical function f(x) in C programming that will be used to demonstrate derivative approximation. We'll create a simple quadratic function to illustrate the concept.

First, let's create a new C file in the ~/project directory:

cd ~/project
nano derivative_approximation.c

Now, let's write the initial code for our function:

#include <stdio.h>
#include <math.h>

// Define a quadratic function f(x) = x^2 + 2x + 1
double f(double x) {
    return x * x + 2 * x + 1;
}

int main() {
    double x = 2.0;  // Point at which we'll approximate the derivative
    printf("Function f(x) = x^2 + 2x + 1\n");
    printf("Evaluated at x = %.2f: f(x) = %.2f\n", x, f(x));

    return 0;
}

Let's compile and run the code to verify our function:

gcc derivative_approximation.c -o derivative_approximation -lm
./derivative_approximation

Example output:

Function f(x) = x^2 + 2x + 1
Evaluated at x = 2.00: f(x) = 9.00

Code Explanation

• We define a quadratic function f(x) = x^2 + 2x + 1
• The function takes a double input x and returns a double result
• In the main() function, we demonstrate evaluating the function at x = 2
• We use printf() to display the function details and its value

Use a Small h and Compute (f(x+h)-f(x))/h

In this step, we will modify our previous code to approximate the derivative using the difference method. We'll introduce a small value h to calculate the numerical derivative.

Let's update the derivative_approximation.c file:

nano ~/project/derivative_approximation.c

Replace the previous main() function with the following code:

#include <stdio.h>
#include <math.h>

// Quadratic function f(x) = x^2 + 2x + 1
double f(double x) {
    return x * x + 2 * x + 1;
}

// Approximate derivative using difference method
double approximate_derivative(double x, double h) {
    return (f(x + h) - f(x)) / h;
}

int main() {
    double x = 2.0;  // Point of derivative approximation
    double h = 0.0001;  // Small value for difference method

    double approx_derivative = approximate_derivative(x, h);

    printf("Function: f(x) = x^2 + 2x + 1\n");
    printf("Approximating derivative at x = %.2f\n", x);
    printf("Step size h = %.6f\n", h);
    printf("Approximate derivative: %.4f\n", approx_derivative);

    return 0;
}

Compile and run the updated code:

gcc derivative_approximation.c -o derivative_approximation -lm
./derivative_approximation

Example output:

Function: f(x) = x^2 + 2x + 1
Approximating derivative at x = 2.00
Step size h = 0.000100
Approximate derivative: 5.0001

Code Explanation

• We introduce a new function approximate_derivative() that calculates the derivative using the difference method
• h is a small value (0.0001) that helps approximate the instantaneous rate of change
• The formula (f(x+h) - f(x)) / h approximates the derivative at point x
• We print the approximated derivative value

Print the Approximate Derivative

In this step, we will expand our derivative approximation program to compare the numerical approximation with the analytical derivative and print the results in a more informative way.

Update the derivative_approximation.c file:

nano ~/project/derivative_approximation.c

Replace the previous code with the following:

#include <stdio.h>
#include <math.h>

// Quadratic function f(x) = x^2 + 2x + 1
double f(double x) {
    return x * x + 2 * x + 1;
}

// Analytical derivative of f(x)
double analytical_derivative(double x) {
    return 2 * x + 2;
}

// Approximate derivative using difference method
double approximate_derivative(double x, double h) {
    return (f(x + h) - f(x)) / h;
}

int main() {
    double x = 2.0;  // Point of derivative approximation
    double h_values[] = {1e-1, 1e-2, 1e-3, 1e-4, 1e-5};
    int num_h = sizeof(h_values) / sizeof(h_values[0]);

    double true_derivative = analytical_derivative(x);

    printf("Function: f(x) = x^2 + 2x + 1\n");
    printf("Derivative Point: x = %.2f\n", x);
    printf("Analytical Derivative: %.4f\n\n", true_derivative);

    printf("Derivative Approximation Results:\n");
    printf("-----------------------------------\n");
    printf("Step Size (h)   Approximate Derivative   Error\n");
    printf("-----------------------------------\n");

    for (int i = 0; i < num_h; i++) {
        double h = h_values[i];
        double approx_derivative = approximate_derivative(x, h);
        double error = fabs(true_derivative - approx_derivative);

        printf("%.1e             %.4f               %.6f\n",
               h, approx_derivative, error);
    }

    return 0;
}

Compile and run the updated code:

gcc derivative_approximation.c -o derivative_approximation -lm
./derivative_approximation

Example output:

Function: f(x) = x^2 + 2x + 1
Derivative Point: x = 2.00
Analytical Derivative: 6.0000

Derivative Approximation Results:
-----------------------------------
Step Size (h)   Approximate Derivative   Error
-----------------------------------
1.0e-01             6.200000               0.200000
1.0e-02             6.020000               0.020000
1.0e-03             6.002000               0.002000
1.0e-04             6.000200               0.000200
1.0e-05             6.000020               0.000020

Code Explanation

• Added analytical_derivative() function to calculate the true derivative
• Created an array of different step sizes h to demonstrate convergence
• Used a loop to print approximations with different step sizes
• Calculated and displayed the error between analytical and numerical derivatives
• Demonstrates how smaller h values lead to more accurate approximations

Summary

In this lab, we first defined a quadratic function f(x) = x^2 + 2x + 1 in C programming. We then introduced a small value h to approximate the derivative of the function using the difference method, (f(x+h)-f(x))/h. Finally, we printed the approximate derivative value.