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 (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 2: Differentiate the inner function and simplify. (dU)/(dx) = 2(2x+3) · (2) = 4(2x+3) = 8x + 12 The derivative is 8x + 12. b) V = (3x - 1)^2/3 Step 1: Apply the chain rule. Let u = 3x-1. (dV)/(dx) = (2)/(3)(3x-1)^(2)/(3)-1 · (d)/(dx)(3x-1) Step 2: Differentiate the inner function and simplify. (dV)/(dx) = (2)/(3)(3x-1)^-(1)/(3) · (3) = 2(3x-1)^-(1)/(3) Step 3: Rewrite with positive exponents. (dV)/(dx) = (2)/((3x-1)^1/3) The derivative is (2)/((3x-1)^1/3). c) y = (1)/(3x+2) Step 1: Rewrite the function using negative exponents. y = (3x+2)^-1 Step 2: Apply the chain rule. Let u = 3x+2. (dy)/(dx) = -1(3x+2)^-1-1 · (d)/(dx)(3x+2) Step 3: Differentiate the inner function and simplify. (dy)/(dx) = -1(3x+2)^-2 · (3) = -3(3x+2)^-2 Step 4: Rewrite with positive exponents. (dy)/(dx) = -(3)/((3x+2)^2) The derivative is -(3)/((3x+2)^2). d) y = (1)/((3x+2)^2) Step 1: Rewrite the function using negative exponents. y = (3x+2)^-2 Step 2: Apply the chain rule. Let u = 3x+2. (dy)/(dx) = -2(3x+2)^-2-1 · (d)/(dx)(3x+2) Step 3: Differentiate the inner function and simplify. (dy)/(dx) = -2(3x+2)^-3 · (3) = -6(3x+2)^-3 Step 4: Rewrite with positive exponents. (dy)/(dx) = -(6)/((3x+2)^3) The derivative is -(6)/((3x+2)^3). e) y = (1)/(sqrt(3x^2+2x-1)) Step 1: Rewrite the function using fractional and negative exponents. y = (3x^2+2x-1)^-1/2 Step 2: Apply the chain rule. Let u = 3x^2+2x-1. (dy)/(dx) = -(1)/(2)(3x^2+2x-1)^-(1)/(2)-1 · (d)/(dx)(3x^2+2x-1) Step 3: Differentiate the inner function. (d)/(dx)(3x^2+2x-1) = 6x + 2 Step 4: Substitute and simplify. (dy)/(dx) = -(1)/(2)(3x^2+2x-1)^-(3)/(2) · (6x+2) = -(3x+1)(3x^2+2x-1)^-(3)/(2) Step 5: Rewrite with positive exponents. (dy)/(dx) = -(3x+1)/((3x^2+2x-1)^3/2) The derivative is -(3x+1)/((3x^2+2x-1)^3/2). f) y = sqrt(3x^2+2x-1) Step 1: Rewrite the function using fractional exponents. y = (3x^2+2x-1)^1/2 Step 2: Apply the chain rule. Let u = 3x^2+2x-1. (dy)/(dx) = (1)/(2)(3x^2+2x-1)^(1)/(2)-1 · (d)/(dx)(3x^2+2x-1) Step 3: Differentiate the inner function. (d)/(dx)(3x^2+2x-1) = 6x + 2 Step 4: Substitute and simplify. (dy)/(dx) = (1)/(2)(3x^2+2x-1)^-(1)/(2) · (6x+2) = (3x+1)(3x^2+2x-1)^-(1)/(2) Step 5: Rewrite with positive exponents. (dy)/(dx) = (3x+1)/(sqrt(3x^2+2x-1)) The derivative is (3x+1)/(sqrt(3x^2+2x-1)). g) y = (3sqrt(x) - 2x)^3 Step 1: Rewrite the function with fractional exponents for the inner term. y = (3x^1/2 - 2x)^3 Step 2: Apply the chain rule. Let u = 3x^1/2 - 2x. (dy)/(dx) = 3(3x^1/2 - 2x)^3-1 · (d)/(dx)(3x^1/2 - 2x) Step 3: Differentiate the inner function. (d)/(dx)(3x^1/2 - 2x) = 3 · (1)/(2)x^(1)/(2)-1 - 2 = (3)/(2)x^-(1)/(2) - 2 Step 4: Substitute and simplify. (dy)/(dx) = 3(3x^1/2 - 2x)^2 ((3)/(2)x^-1/2 - 2) (dy)/(dx) = 3(3sqrt(x) - 2x)^2 ((3)/(2sqrt(x)) - 2) The derivative is 3(3sqrt(x) - 2x)^2 ((3)/(2sqrt(x)) - 2). That's 2 down. 3 left today — send the next one.