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 differentials for each function: a) y = 4x^3 - 2x + 5 Step 1: Find the derivative (dy)/(dx). (dy)/(dx) = (d)/(dx)(4x^3 - 2x + 5) = 4(3x^3-1) - 2(1) + 0 = 12x^2 - 2 Step 2: Write the differential dy. dy = (12x^2 - 2) \, dx The differential is dy = (12x^2 - 2) \, dx. b) y = 2sqrt(x^4 + 6x) Step 1: Rewrite the function using fractional exponents. y = 2(x^4 + 6x)^1/2 Step 2: Find the derivative (dy)/(dx) using the chain rule. (dy)/(dx) = 2 · (1)/(2)(x^4 + 6x)^(1/2)-1 · (d)/(dx)(x^4 + 6x) (dy)/(dx) = (x^4 + 6x)^-1/2 · (4x^3 + 6) (dy)/(dx) = (4x^3 + 6)/(sqrt(x^4 + 6x)) Step 3: Write the differential dy. dy = (4x^3 + 6)/(sqrt(x^4 + 6x)) \, dx The differential is dy = (4x^3 + 6)/(sqrt(x^4 + 6x)) \, dx. c) y = (x^3 - 5x + 11) Step 1: Find the derivative (dy)/(dx) using the chain rule. Let u = x^3 - 5x + 11. Then y = u. (dy)/(dx) = (d)/(du)( u) · (du)/(dx) (dy)/(dx) = (- u) · (d)/(dx)(x^3 - 5x + 11) (dy)/(dx) = -(x^3 - 5x + 11) · (3x^2 - 5) (dy)/(dx) = -(3x^2 - 5)(x^3 - 5x + 11) Step 2: Write the differential dy. dy = -(3x^2 - 5)(x^3 - 5x + 11) \, dx The differential is dy = -(3x^2 - 5)(x^3 - 5x + 11) \, dx. d) y = (x^10 + (2x)) Step 1: Find the derivative (dy)/(dx). (dy)/(dx) = (d)/(dx)(x^10) + (d)/(dx)((2x)) For (d)/(dx)((2x)), use the chain rule. Let u = 2x, then (d)/(dx)( u) = u · (du)/(dx) = (2x) · 2 = 2(2x). (dy)/(dx) = 10x^9 + 2(2x) Step 2: Write the differential dy. dy = (10x^9 + 2(2x)) \, dx The differential is dy = (10x^9 + 2(2x)) \, dx. 3 done, 2 left today. You're making progress.