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.
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Here are the solutions to the problems: 8.2.1 To determine g'(x), we apply the power rule for differentiation, which states that (d)/(dx)(ax^n) = nax^n-1. Given g(x) = -3x^4 + 2x. Step 1: Differentiate each term in the expression. g'(x) = (d)/(dx)(-3x^4) + (d)/(dx)(2x) Step 2: Apply the power rule to each term. For the first term, (d)/(dx)(-3x^4): -3 · 4 · x^4-1 = -12x^3 For the second term, (d)/(dx)(2x): 2 · 1 · x^1-1 = 2x^0 = 2 · 1 = 2 Step 3: Combine the results. g'(x) = -12x^3 + 2 The derivative is -12x^3 + 2. 8.2.2 To determine (dy)/(dx), we first simplify the expression for y by dividing each term in the numerator by the denominator, then apply the power rule for differentiation. Given y = (2x^4 + 1)/(x^2). Step 1: Rewrite y by separating the terms and simplifying exponents. y = (2x^4)/(x^2) + (1)/(x^2) y = 2x^4-2 + x^-2 y = 2x^2 + x^-2 Step 2: Differentiate each term using the power rule (d)/(dx)(ax^n) = nax^n-1. For the first term, 2x^2: (d)/(dx)(2x^2) = 2 · 2 · x^2-1 = 4x^1 = 4x For the second term, x^-2: (d)/(dx)(x^-2) = -2 · x^-2-1 = -2x^-3 Step 3: Combine the results to find (dy)/(dx). (dy)/(dx) = 4x - 2x^-3 The derivative is 4x - 2x^-3.