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
Hey mercyjoe8687 — ready when you are. Here are the solutions using the chain rule: Basic Polynomial Composition: 1. To find (dy)/(dx) for y = (3x + 2)^5: (dy)/(dx) = 5(3x + 2)^5-1 · (d)/(dx)(3x + 2) (dy)/(dx) = 5(3x + 2)^4 · 3 15(3x + 2)^4 2. To differentiate f(x) = (2x^2 - 5x + 1)^4: f'(x) = 4(2x^2 - 5x + 1)^4-1 · (d)/(dx)(2x^2 - 5x + 1) f'(x) = 4(2x^2 - 5x + 1)^3 · (4x - 5) 4(4x - 5)(2x^2 - 5x + 1)^3 3. To find f'(x) if f(x) = (4 - x^2)^3: f'(x) = 3(4 - x^2)^3-1 · (d)/(dx)(4 - x^2) f'(x) = 3(4 - x^2)^2 · (-2x) -6x(4 - x^2)^2 Square Root and Fractional Powers: 4. To differentiate y = sqrt(5x^2 + 3x - 1) = (5x^2 + 3x - 1)^1/2: (dy)/(dx) = (1)/(2)(5x^2 + 3x - 1)^(1)/(2)-1 · (d)/(dx)(5x^2 + 3x - 1) (dy)/(dx) = (1)/(2)(5x^2 + 3x - 1)^-1/2 · (10x + 3) (10x + 3)/(2sqrt(5x^2 + 3x - 1)) 5. To find (dy)/(dx) if y = (x^3 + 2x)^1/3: (dy)/(dx) = (1)/(3)(x^3 + 2x)^(1)/(3)-1 · (d)/(dx)(x^3 + 2x) (dy)/(dx) = (1)/(3)(x^3 + 2x)^-2/3 · (3x^2 + 2) (3x^2 + 2)/(3(x^3 + 2x)^2/3) 6. To find the derivative of f(x) = (1 - 2x)^-2: f'(x) = -2(1 - 2x)^-2-1 · (d)/(dx)(1 - 2x) f'(x) = -2(1 - 2x)^-3 · (-2) f'(x) = 4(1 - 2x)^-3 (4)/((1 - 2x)^3) Trigonometric Functions: 7. To differentiate y = (4x^2 + 3x): (dy)/(dx) = (4x^2 + 3x) · (d)/(dx)(4x^2 + 3x) (dy)/(dx) = (4x^2 + 3x) · (8x + 3) (8x + 3)(4x^2 + 3x) 8. To find f'(x) if f(x) = ^3(2x) = ((2x))^3: f'(x) = 3((2x))^3-1 · (d)/(dx)((2x)) f'(x) = 3^2(2x) · (-(2x)) · (d)/(dx)(2x) f'(x) = 3^2(2x) · (-(2x)) · 2 -6^2(2x)(2x) 9. To find (dy)/(dx) if y = (x^2 - 5x): (dy)/(dx) = ^2(x^2 - 5x) · (d)/(dx)(x^2 - 5x) (dy)/(dx) = ^2(x^2 - 5x) · (2x - 5) (2x - 5)^2(x^2 - 5x) 10. To differentiate y = ^2(3x + /4) = ((3x + /4))^2: (dy)/(dx) = 2(3x + /4) · (d)/(dx)((3x + /4)) (dy)/(dx) = 2(3x + /4) · (3x + /4) · (d)/(dx)(3x + /4) (dy)/(dx) = 2(3x + /4) · (3x + /4) · 3 6(3x + /4)(3x + /4) Exponential Functions: 11. To find (dy)/(dx) if y = e^3x^2 + 2x: (dy)/(dx) = e^3x^2 + 2x · (d)/(dx)(3x^2 + 2x) (dy)/(dx) = e^3x^2 + 2x · (6x + 2) (6x + 2)e^3x^2 +