Give a much shorter, concise version — just the essential steps and the final answer, no padding. Problem: What is the derivative of x3 + 2x with respect to x? Give a full worked solution with steps. Current solution: The derivative of x3 + 2x with respect to x is 3x2 + 2. Full Worked Solution: To find the derivative of f(x) = x3 + 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)/(d x)[x^n] = nx^n-1 3. Constant Multiple Rule: (d)/(dx)[cf(x)] = c(d)/(dx)[f(x)] Let f (x) = x3 + 2x. Step 1: Apply the Sum Rule We can differentiate each term separately: (d)/(dx)(x3 + 2x) = ddx(x3) + (d)/(dx)(2x) Step 2: Differentiate the first term, x3 Using the Power Rule where n=3: (d)/(dx)(x3) = 3x3-1 = 3x2 Step 3: Differentiate the second term, 2x Using the Constant Multiple Rule and the Power Rule (where x = x1, so n=1): (d)/(dx)(2x) = 2 (d)/(dx)(x1) (d)/(dx)(x1) = 1x1-1 = 1x0 = 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)/(d x)(x3 + 2x) = 3x2 + 2

The derivative of x^3 + 2x with respect to x is 3x^2 + 2. Worked Solution: 1. Apply the sum rule: (d)/(dx)(x^3 + 2x) = (d)/(dx)(x^3) + (d)/(dx)(2x) 2. Differentiate x^3 using the power rule: (d)/(dx)(x^3) = 3x^3-1 = 3x^2 3. Differentiate 2x using the constant multiple and power rules: (d)/(dx)(2x) = 2 · (d)/(dx)(x^1) = 2 · 1x^1-1 = 2 · 1 = 2 4. Combine the results: 3x^2 + 2