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 are the differentiations for the given functions. Step 1: Differentiate y = x^5. Using the power rule (d)/(dx)(x^n) = nx^n-1: (dy)/(dx) = 5x^5-1 (dy)/(dx) = 5x^4 (dy)/(dx) = 5x^4 Step 2: Differentiate y = 2x^3. Using the constant multiple rule (d)/(dx)(cf(x)) = c(d)/(dx)(f(x)) and the power rule: (dy)/(dx) = 2 · (3x^3-1) (dy)/(dx) = 2 · 3x^2 (dy)/(dx) = 6x^2 (dy)/(dx) = 6x^2 Step 3: Differentiate y = 3x^2 + 2x + 5. Using the sum rule (d)/(dx)(f(x) + g(x)) = (d)/(dx)(f(x)) + (d)/(dx)(g(x)) and the power rule, and the rule for a constant (d)/(dx)(c) = 0: (dy)/(dx) = (d)/(dx)(3x^2) + (d)/(dx)(2x) + (d)/(dx)(5) (dy)/(dx) = 3 · (2x^2-1) + 2 · (1x^1-1) + 0 (dy)/(dx) = 6x^1 + 2x^0 (dy)/(dx) = 6x + 2 · 1 (dy)/(dx) = 6x + 2 (dy)/(dx) = 6x + 2 That's 2 down. 3 left today — send the next one.