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: 8.1.1 To determine the derivative of the expression, we apply the power rule for differentiation, which states that (d)/(dx)(ax^n) = nax^n-1. Step 1: Differentiate each term in the expression. (d)/(dx) [3x - 5x^2] = (d)/(dx)(3x) - (d)/(dx)(5x^2) Step 2: Apply the power rule to each term. For the first term, (d)/(dx)(3x): (d)/(dx)(3x) = 3 · 1 · x^1-1 = 3x^0 = 3 · 1 = 3 For the second term, (d)/(dx)(5x^2): (d)/(dx)(5x^2) = 5 · 2 · x^2-1 = 10x^1 = 10x Step 3: Combine the results. (d)/(dx) [3x - 5x^2] = 3 - 10x The derivative is 3 - 10x. 8.1.2 To determine g'(x), we first rewrite g(x) using exponent notation, then apply the power rule for differentiation. Step 1: Rewrite g(x) using negative and fractional exponents. Recall that (1)/(x^n) = x^-n and [m]x^n = x^n/m. g(x) = (2)/(x^2) - [3]x^7 g(x) = 2x^-2 - x^7/3 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^-3 For the second term, -x^7/3: (d)/(dx)(-x^7/3) = -(7)/(3) · x^7/3-1 = -(7)/(3)x^7/3-3/3 = -(7)/(3)x^4/3 Step 3: Combine the results to find g'(x). g'(x) = -4x^-3 - (7)/(3)x^4/3 The derivative is -4x^-3 - (7)/(3)x^4/3.