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: 6.1 To determine the derivative f'(x) of f(x) = -2x^2 from first principles, we use the definition: f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 1: Find f(x+h). f(x+h) = -2(x+h)^2 = -2(x^2 + 2xh + h^2) = -2x^2 - 4xh - 2h^2 Step 2: Find the difference f(x+h) - f(x). f(x+h) - f(x) = (-2x^2 - 4xh - 2h^2) - (-2x^2) f(x+h) - f(x) = -2x^2 - 4xh - 2h^2 + 2x^2 f(x+h) - f(x) = -4xh - 2h^2 Step 3: Form the difference quotient (f(x+h) - f(x))/(h). (f(x+h) - f(x))/(h) = (-4xh - 2h^2)/(h) = (h(-4x - 2h))/(h) = -4x - 2h Step 4: Take the limit as h 0. f'(x) = _h 0 (-4x - 2h) = -4x - 2(0) = -4x The derivative is -4x. 6.2.1 To differentiate f(x) = 3x^2 - 6x with respect to x, we apply the power rule (d)/(dx)(ax^n) = nax^n-1. Step 1: Differentiate each term. f'(x) = (d)/(dx)(3x^2) - (d)/(dx)(6x) Step 2: Apply the power rule. (d)/(dx)(3x^2) = 3 · 2x^2-1 = 6x (d)/(dx)(6x) = 6 · 1x^1-1 = 6x^0 = 6 Step 3: Combine the results. f'(x) = 6x - 6 The derivative is 6x - 6. 6.2.2 To determine (dy)/(dx) if y = 2sqrt(x) - (8)/(x), we first rewrite y using exponent notation. Step 1: Rewrite y using exponents. y = 2x^1/2 - 8x^-1 Step 2: Differentiate each term using the power rule. (dy)/(dx) = (d)/(dx)(2x^1/2) - (d)/(dx)(8x^-1) (d)/(dx)(2x^1/2) = 2 · (1)/(2) x^1/2 - 1 = x^-1/2 (d)/(dx)(8x^-1) = 8 · (-1) x^-1 - 1 = -8x^-2 Step 3: Combine the results. (dy)/(dx) = x^-1/2 - (-8x^-2) = x^-1/2 + 8x^-2 The derivative is x^-1/2 + 8x^-2. 6.3 To determine (dy)/(dx) if xy - 5 = sqrt(x^3), we first solve for y explicitly. Step 1: Isolate y. xy = sqrt(x^3) + 5 xy = x^3/2 + 5 y = x^3/2 + 5x y = x^3/2x + (5)/(x) y = x^3/2 - 1 + 5x^-1 y = x^1/2 + 5x^-1 Step 2: Differentiate y with respect to x using the power rule. (dy)/(dx) = (d)/(dx)(x^1/2) + (d)/(dx)(5x^-1) (d)/(dx)(x^1/2) = (1)/(2)x^1/2 - 1 = (1)/(2)x^-1/2 (d)/(dx)(5x^-1) = 5 · (-1)x^-1 - 1 = -5x^-2 Step 3: Combine the results. (dy)/(dx) = (1)/(2)x^-1/2 - 5x^-2 The derivative is (1)/(2)x^-1/2 - 5x^-2. 6.4 To determine the equation of the tangent to the graph of f(x) = 10x - x^2 at the point where f'(x) = -2. Step 1: Find the derivative f'(x). f(x) = 10x - x^2 f'(x) = (d)/(dx)(10x) - (d)/(dx)(x^2) = 10 - 2x Step 2: Find the x-coordinate of the point of tangency by setting f'(x) = -2. 10 - 2x = -2 -2x = -2 - 10 -2x = -12 x = 6 Step 3: Find the y-coordinate of the point of tangency by substituting x=6 into f(x). f(6) = 10(6) - (6)^2 f(6) = 60 - 36 f(6) = 24 The point of tangency is (6, 24). The slope of the tangent (m) is given as -2. Step 4: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1). y - 24 = -2(x - 6) y - 24 = -2x + 12 y = -2x + 12 + 24 y = -2x + 36 The equation of the tangent is y = -2x + 36.