Round #1000

Differentiate f(x) = (3x^2 + 1)^4 with respect to x.

To differentiate f(x) = (3x^2 + 1)^4 with respect to x, we'll use the chain rule. The chain rule states that if we have a composite function of the form f(g(x)), then the derivative of f with respect to x is given by: f'(x) = f'(g(x)) * g'(x) In this case, we have: f(x) = (3x^2 + 1)^4 g(x) = 3x^2 + 1 First, let's find the derivative of g(x) with respect to x: g'(x) = d(3x^2 + 1)/dx = 6x Now, let's find the derivative of f with respect to g(x): f'(g(x)) = d((3x^2 + 1)^4)/d(3x^2 + 1) = 4(3x^2 + 1)^3 Now, we can apply the chain rule: f'(x) = f'(g(x)) * g'(x) = 4(3x^2 + 1)^3 * 6x = 24x(3x^2 + 1)^3 Therefore, the derivative of f(x) = (3x^2 + 1)^4 with respect to x is: f'(x) = 24x(3x^2 + 1)^3 Here is the C# code to calculate the derivative: ```csharp using System; public class Program { public static void Main() { Console.Write("Enter the value of x: "); double x = Convert.ToDouble(Console.ReadLine()); double derivative = CalculateDerivative(x); Console.WriteLine("The derivative of f(x) = (3x^2 + 1)^4 at x = " + x + " is " + derivative); } public static double CalculateDerivative(double x) { return 24 * x * Math.Pow(3 * x * x + 1, 3); } } ``` This code will calculate the derivative of the given function at a specified value of x.

To differentiate f(x) = (3x^2 + 1)^4 with respect to x, we can apply the chain rule. The chain rule states that if we have a composite function of the form f(g(x)), then its derivative is given by f'(g(x)) * g'(x). First, let's identify the outer and inner functions. The outer function is u^4, where u = 3x^2 + 1. The derivative of the outer function with respect to u is 4u^3. The derivative of the inner function u = 3x^2 + 1 with respect to x is 6x. Now, we can apply the chain rule: f'(x) = d( (3x^2 + 1)^4 )/dx = 4(3x^2 + 1)^3 * d(3x^2 + 1)/dx = 4(3x^2 + 1)^3 * 6x = 24x(3x^2 + 1)^3 This is the derivative of f(x) = (3x^2 + 1)^4 with respect to x. Here is the C# code to calculate the derivative: ```csharp using System; public class Program { public static void Main() { // Define the function and its derivative Func<double, double> f = x => Math.Pow(3 * Math.Pow(x, 2) + 1, 4); Func<double, double> fDerivative = x => 24 * x * Math.Pow(3 * Math.Pow(x, 2) + 1, 3); // Test the derivative at a point double x = 1.0; Console.WriteLine($"f({x}) = {f(x)}"); Console.WriteLine($"f'({x}) = {fDerivative(x)}"); } } ``` This code defines the function f(x) and its derivative f'(x), and tests the derivative at a point x = 1.0. The output will be the value of the function and its derivative at the given point.