Round #1077

Differentiate f(x) = 7x^4 - 2x^3 + x - 9. Give the derivative f'(x) as a polynomial.

To find the derivative of f(x) = 7x^4 - 2x^3 + x - 9, 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 7x^4: - Using the power rule, the derivative of x^4 is 4x^(4-1) = 4x^3. - Since the term is 7x^4, we multiply the derivative by 7, resulting in 7*4x^3 = 28x^3. 2. Differentiate -2x^3: - Using the power rule, the derivative of x^3 is 3x^(3-1) = 3x^2. - Since the term is -2x^3, we multiply the derivative by -2, resulting in -2*3x^2 = -6x^2. 3. Differentiate x: - Using the power rule, the derivative of x (which is x^1) is 1x^(1-1) = 1x^0 = 1. 4. Differentiate -9: - The derivative of a constant is always 0, so the derivative of -9 is 0. Now, combine the derivatives of each term to get the derivative of the entire function: f'(x) = 28x^3 - 6x^2 + 1 So, the derivative of f(x) = 7x^4 - 2x^3 + x - 9 is f'(x) = 28x^3 - 6x^2 + 1.

```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 = "7x^4 - 2x^3 + x - 9"; 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. The derivative of f(x) = 7x^4 - 2x^3 + x - 9 is 28x^3 - 6x^2 + 1.