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.
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The derivative of x^3 + 2x with respect to x is 3x^2 + 2. Full Worked Solution: To find the derivative of f(x) = x^3 + 2x, we will use the following differentiation rules: 1. Sum Rule: (d)/(dx)[g(x) + h(x)] = (d)/(dx)[g(x)] + (d)/(dx)[h(x)] 2. Power Rule: (d)/(dx)[x^n] = nx^n-1 3. Constant Multiple Rule: (d)/(dx)[cf(x)] = c(d)/(dx)[f(x)] Let f(x) = x^3 + 2x. Step 1: Apply the Sum Rule We can differentiate each term separately: (d)/(dx)(x^3 + 2x) = (d)/(dx)(x^3) + (d)/(dx)(2x) Step 2: Differentiate the first term, x^3 Using the Power Rule where n=3: (d)/(dx)(x^3) = 3x^3-1 = 3x^2 Step 3: Differentiate the second term, 2x Using the Constant Multiple Rule and the Power Rule (where x = x^1, so n=1): (d)/(dx)(2x) = 2 · (d)/(dx)(x^1) (d)/(dx)(x^1) = 1x^1-1 = 1x^0 = 1 · 1 = 1 So, (d)/(dx)(2x) = 2 · 1 = 2 Step 4: Combine the results Substitute the derivatives of each term back into the expression from Step 1: (d)/(dx)(x^3 + 2x) = 3x^2 + 2