Round #1049

Differentiate g(x) = sin(3x^2 + 1) using the chain rule. Give the exact derivative.

To differentiate g(x) = sin(3x^2 + 1) using the chain rule, we'll follow these steps: 1. Identify the outer function and the inner function. - Outer function: sin(u) - Inner function: u = 3x^2 + 1 2. Differentiate the outer function with respect to u. - The derivative of sin(u) with respect to u is cos(u). 3. Differentiate the inner function with respect to x. - The derivative of 3x^2 + 1 with respect to x is 6x. 4. Apply the chain rule by multiplying the derivative of the outer function (with respect to u) by the derivative of the inner function (with respect to x). - This gives us cos(u) * 6x. 5. Substitute back u = 3x^2 + 1 into the equation from step 4 to express the derivative in terms of x. - This gives us cos(3x^2 + 1) * 6x. So, the derivative g'(x) of g(x) = sin(3x^2 + 1) is: g'(x) = 6x * cos(3x^2 + 1) This is the exact derivative of the given function.

To differentiate g(x) = sin(3x^2 + 1) using the chain rule, we apply the formula (f∘g)'(x) = f'(g(x))·g'(x). Here, f(u) = sin(u) and g(x) = 3x^2 + 1. First, find the derivatives of f(u) and g(x): - f'(u) = cos(u) - g'(x) = d(3x^2 + 1)/dx = 6x (using the power rule) Now, substitute g(x) into f'(u) and multiply by g'(x): g'(x) = f'(g(x))·g'(x) = cos(3x^2 + 1)·6x So, the exact derivative of g(x) = sin(3x^2 + 1) is g'(x) = 6x·cos(3x^2 + 1). Here is the implementation in C#: ```csharp using System; public class Program { public static void Main() { // Define the function g(x) = sin(3x^2 + 1) Func<double, double> g = x => Math.Sin(3 * Math.Pow(x, 2) + 1); // Define the derivative of g(x) using the chain rule Func<double, double> gDerivative = x => 6 * x * Math.Cos(3 * Math.Pow(x, 2) + 1); // Test the derivative at a point, e.g., x = 1 double x = 1; double derivativeAtX = gDerivative(x); Console.WriteLine($"The derivative of g(x) at x = {x} is {derivativeAtX}"); } } ``` This code defines the function g(x) and its derivative using the chain rule, then tests the derivative at a point. The derivative is calculated using the formula derived above: g'(x) = 6x·cos(3x^2 + 1).