This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's how to find the derivative of f(x) = (x^3 + 3x): Step 1: Apply the chain rule. The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) · h'(x). In this case, the outer function is g(u) = (u) and the inner function is h(x) = x^3 + 3x. Step 2: Differentiate the outer function with respect to its argument. The derivative of (u) is (1)/(u). So, g'(u) = (1)/(u). Step 3: Differentiate the inner function with respect to x. The derivative of x^3 + 3x is 3x^2 + 3. So, h'(x) = 3x^2 + 3. Step 4: Combine the derivatives using the chain rule. Substitute u = x^3 + 3x back into g'(u) and multiply by h'(x). f'(x) = (1)/(x^3 + 3x) · (3x^2 + 3) Step 5: Simplify the expression. f'(x) = (3x^2 + 3)/(x^3 + 3x) We can factor out 3 from the numerator and x from the denominator: f'(x) = (3(x^2 + 1))/(x(x^2 + 3)) The final answer is (3(x^2 + 1))/(x(x^2 + 3)). 3 done, 2 left today. You're making progress.