To evaluate (dy)/(dx) if y = (8 - 3x6)/(8x5):

Here are the solutions to the differentiation problems. 1.1 To evaluate (dy)/(dx) if y = (8 - 3x^6)/(8x^5): Step 1: Rewrite the function by separating the terms and expressing them with negative exponents. y = (8)/(8x^5) - (3x^6)/(8x^5) = x^-5 - (3)/(8)x^6-5 = x^-5 - (3)/(8)x Step 2: Differentiate y with respect to x using the power rule, (d)/(dx)(x^n) = nx^n-1. (dy)/(dx) = -5x^-5-1 - (3)/(8)(1)x^1-1 = -5x^-6 - (3)/(8) Step 3: Express the answer with positive exponents. (dy)/(dx) = -(5)/(x^6) - (3)/(8) The final answer is -(5)/(x^6) - (3)/(8). 1.2 To evaluate (dy)/(dx) if y = 2sqrt(x) + 1x^2: Step 1: Rewrite the function by separating the terms and expressing them with fractional and negative exponents. y = 2x^1/2x^2 + (1)/(x^2) = 2x^1/2 - 2 + x^-2 = 2x^-3/2 + x^-2 Step 2: Differentiate y with respect to x using the power rule. (dy)/(dx) = 2(-(3)/(2))x^-3/2 - 1 + (-2)x^-2 - 1 = -3x^-5/2 - 2x^-3 Step 3: Express the answer with positive exponents. (dy)/(dx) = -(3)/(x^5/2) - (2)/(x^3) The final answer is -(3)/(x^5/2) - (2)/(x^3). 1.3 To evaluate (dy)/(dw) if y = 4([3]x^2) and x = w^-3: Step 1: Rewrite y in terms of x with fractional exponents and find (dy)/(dx). y = 4x^2/3 (dy)/(dx) = 4 · (2)/(3)x^2/3 - 1 = (8)/(3)x^-1/3 Step 2: Find (dx)/(dw). x = w^-3 (dx)/(dw) = -3w^-3-1 = -3w^-4 Step 3: Use the chain rule (dy)/(dw) = (dy)/(dx) · (dx)/(dw). Substitute x = w^-3 into (dy)/(dx). (dy)/(dx) = (8)/(3)(w^-3)^-1/3 = (8)/(3)w^(-3)(-1/3) = (8)/(3)w^1 = (8)/(3)w Step 4: Multiply (dy)/(dx) and (dx)/(dw). (dy)/(dw) = ((8)/(3)w)(-3w^-4) = -8w^1-4 = -8w^-3 Step 5: Express the answer with positive exponents. (dy)/(dw) = -(8)/(w^3) The final answer is -(8)/(w^3). 1.4 To evaluate f'(x) if f(x) = (x^3 - 5x^2 + 4x)/(x-4): Step 1: Factor the numerator of f(x). x^3 - 5x^2 + 4x = x(x^2 - 5x + 4) = x(x-1)(x-4) Step 2: Simplify f(x) by canceling common factors. f(x) = (x(x-1)(x-4))/(x-4) = x(x-1) = x^2 - x for x ≠ 4 Step 3: Differentiate f(x) with respect to x using the power rule. f'(x) = (d)/(dx)(x^2 - x) = 2x - 1 The final answer is 2x - 1. 1.5 To evaluate f'(x) if f(x) = (x - (3)/(x))^2: Step 1: Rewrite the function with negative exponents and expand the square. f(x) = (x - 3x^-1)^2 = x^2 - 2(x)(3x^-1) + (3x^-1)^2 f(x) = x^2 - 6x^1-1 + 9x^-2 = x^2 - 6 + 9x^-2 Step 2: Differentiate f(x) with respect to x using the power rule. f'(x) = (d)/(dx)(x^2 - 6 + 9x^-2) = 2x - 0 + 9(-2)x^-2-1 = 2x - 18x^-3 Step 3: Express the answer with positive exponents. f'(x) = 2x - (18)/(x^3) The final answer is 2x - (18)/(x^3). 1.6 To evaluate (dy)/(dx) if x^2 - 3 = [3]y: Step 1: Isolate y by cubing both sides of the equation. y =