Round #1047

Differentiate f(x) = 4x^3 - 5x^2 + 2x - 7. Give the derivative f'(x) as a polynomial.

To find the derivative of f(x) = 4x^3 - 5x^2 + 2x - 7, we'll apply the power rule of differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1). Here's the step-by-step differentiation: 1. Differentiate the first term: The derivative of 4x^3 is 4 * 3 * x^(3-1) = 12x^2. 2. Differentiate the second term: The derivative of -5x^2 is -5 * 2 * x^(2-1) = -10x. 3. Differentiate the third term: The derivative of 2x is 2 * 1 * x^(1-1) = 2. 4. Differentiate the fourth term: The derivative of a constant (in this case, -7) is 0, since the derivative of a constant is always 0. Now, combine the results: f'(x) = 12x^2 - 10x + 2 So, the derivative of f(x) = 4x^3 - 5x^2 + 2x - 7 is f'(x) = 12x^2 - 10x + 2.

```csharp using System; public class PolynomialDerivative { public static string DifferentiatePolynomial(string polynomial) { // Split the polynomial into terms string[] terms = polynomial.Split('+'); // Initialize the derivative string derivative = ""; // Apply the power rule and linearity of differentiation to each term foreach (string term in terms) { // Remove any whitespace term = term.Trim(); // Check if the term is a constant if (!term.Contains("x")) { continue; } // Extract the coefficient and exponent string[] parts = term.Split('x'); int coefficient = 1; if (parts[0] != "") { coefficient = int.Parse(parts[0]); } int exponent = 1; if (parts.Length > 1 && parts[1] != "") { exponent = int.Parse(parts[1].Replace("^", "")); } // Apply the power rule int newCoefficient = coefficient * exponent; int newExponent = exponent - 1; // Add the term to the derivative if (newCoefficient != 0) { if (derivative != "") { derivative += " + "; } if (newCoefficient != 1 || newExponent == 0) { derivative += newCoefficient; } if (newExponent > 0) { derivative += "x"; if (newExponent > 1) { derivative += "^" + newExponent; } } } } // Remove any leading or trailing whitespace derivative = derivative.Trim(); return derivative; } public static void Main() { string polynomial = "4x^3 - 5x^2 + 2x - 7"; polynomial = polynomial.Replace("-", " -"); polynomial = polynomial.Replace(" ", ""); string derivative = DifferentiatePolynomial(polynomial); Console.WriteLine(derivative); } } ``` This implementation uses the power rule and linearity of differentiation to find the derivative of the given polynomial. The `DifferentiatePolynomial` function splits the polynomial into terms, applies the power rule to each term, and combines the results to form the derivative. The `Main` function demonstrates how to use this function with the given polynomial.