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
Right Ni_Fe_Mi✨, let's go. Part (A) a) y = x Step 1: Apply the power rule (d)/(dx)(x^n) = nx^n-1. (dy)/(dx) = 1 · x^1-1 = 1 · x^0 = 1 The derivative is 1. b) y = -x^2 + 3 Step 1: Differentiate each term using the power rule and constant rule. (dy)/(dx) = -2x^2-1 + 0 = -2x The derivative is -2x. c) y = (x^3)/(3) + (x^2)/(2) + x Step 1: Differentiate each term using the power rule. (dy)/(dx) = (1)/(3)(3x^3-1) + (1)/(2)(2x^2-1) + 1 Step 2: Simplify the expression. (dy)/(dx) = x^2 + x + 1 The derivative is x^2 + x + 1. d) y = 3x^7 - 7x^3 + 21x^2 Step 1: Differentiate each term using the power rule. (dy)/(dx) = 3(7x^7-1) - 7(3x^3-1) + 21(2x^2-1) Step 2: Simplify the expression. (dy)/(dx) = 21x^6 - 21x^2 + 42x The derivative is 21x^6 - 21x^2 + 42x. e) y = (x^4)/(4) - (x^3)/(3) + (x^2)/(2) - x + 3 Step 1: Differentiate each term using the power rule and constant rule. (dy)/(dx) = (1)/(4)(4x^4-1) - (1)/(3)(3x^3-1) + (1)/(2)(2x^2-1) - 1 + 0 Step 2: Simplify the expression. (dy)/(dx) = x^3 - x^2 + x - 1 The derivative is x^3 - x^2 + x - 1. f) y = sqrt(x^5) Step 1: Rewrite the function using fractional exponents. y = x^5/2 Step 2: Apply the power rule. (dy)/(dx) = (5)/(2)x^(5)/(2)-1 = (5)/(2)x^(3)/(2) The derivative is (5)/(2)x^3/2. g) y = (1)/(x^2/5) Step 1: Rewrite the function using negative exponents. y = x^-2/5 Step 2: Apply the power rule. (dy)/(dx) = -(2)/(5)x^-(2)/(5)-1 = -(2)/(5)x^-(7)/(5) Step 3: Rewrite with positive exponents. (dy)/(dx) = -(2)/(5x^7/5) The derivative is -(2)/(5x^7/5). Part (B) a) U = (2x + 3)^2 Step 1: Apply the chain rule. Let u = 2x+3. (dU)/(dx) = 2(2x+3)^2-1 · (d)/(dx)(2x+3) Step