Find the derivative of f(x) = -3x2/x3 using first principles, and also use differentiation rules for y = 4e2x + 1/2 ln x - 3/√x + sin 30°.

3.1 To determine the derivative of f(x) = (-3x^2)/(x^3) using first principles, we first simplify f(x). Step 1: Simplify f(x). f(x) = (-3x^2)/(x^3) = -3x^-1 Step 2: Apply the first principles formula f'(x) = _h 0 (f(x+h) - f(x))/(h). f'(x) = _h 0 -3(x+h)^-1 - (-3x^-1)h f'(x) = _h 0 (-3)/(x+h) + (3)/(x)h Step 3: Combine the fractions in the numerator. f'(x) = _h 0 (-3x + 3(x+h))/(x(x+h))h f'(x) = _h 0 (-3x + 3x + 3h)/(x(x+h))h f'(x) = _h 0 (3h)/(x(x+h))h Step 4: Simplify by canceling h. f'(x) = _h 0 (3)/(x(x+h)) Step 5: Evaluate the limit as h 0. f'(x) = (3)/(x(x+0)) = (3)/(x^2) The derivative of f(x) by first principles is (3)/(x^2). 3.2 Use differentiation rules to determine the derivatives of each of the following: 3.2.1 Given y = 4e^2x + (1)/(2) x - (3)/(sqrt(x)) + 30^. Step 1: Rewrite terms with exponents for differentiation. y = 4e^2x + (1)/(2) x - 3x^-1/2 + 30^ Step 2: Differentiate each term. Remember 30^ is a constant. (dy)/(dx) = (d)/(dx)(4e^2x) + (d)/(dx)((1)/(2) x) - (d)/(dx)(3x^-1/2) + (d)/(dx)( 30^) (dy)/(dx) = 4(2e^2x) + (1)/(2)((1)/(x)) - 3(-(1)/(2)x^-3/2) + 0 (dy)/(dx) = 8e^2x + (1)/(2x) + (3)/(2)x^-3/2 Step 3: Express with positive exponents and in surd form. (dy)/(dx) = 8e^2x + (1)/(2x) + (3)/(2sqrt(x^3)) The derivative is 8e^2x + (1)/(2x) + (3)/(2sqrt(x^3)). 3.2.2 Given y = (6x^4 - 3x^2)/(x^2). Step 1: Simplify the expression. y = (6x^4)/(x^2) - (3x^2)/(x^2) = 6x^2 - 3 Step 2: Differentiate the simplified expression. (dy)/(dx) = (d)/(dx)(6x^2 - 3) (dy)/(dx) = 12x - 0 (dy)/(dx) = 12x The derivative is 12x. 3.2.3 Given y = 3x^3 (2x). Step 1: Use the product rule: (d)/(dx)(uv) = u'v + uv'. Let u = 3x^3 and v = (2x). Then u' = 9x^2 and v' = 2(2x). Step 2: Apply the product rule. (dy)/(dx) = (9x^2)((2x)) + (3x^3)(2(2x)) (dy)/(dx) = 9x^2 (2x) + 6x^3 (2x) The derivative is 9x^2 (2x) + 6x^3 (2x). 3.2.4 Given y = (3)/((2-x)^4). Step 1: Rewrite the expression with a negative exponent. y = 3(2-x)^-4 Step 2: Use the chain rule: (dy)/(dx) = (dy)/(du) · (du)/(dx). Let u = 2-x, so (du)/(dx) = -1. Then y = 3u^-4, so (dy)/(du) = 3(-4)u^-5 = -12u^-5. Step 3: Apply the chain rule. (dy)/(dx) = (-12u^-5)(-1) (dy)/(dx) = 12u^-5 Step 4: Substitute back u = 2-x and express with a positive exponent. (dy)/(dx) = 12(2-x)^-5 = (12)/((2-x)^5) The derivative is (12)/((2-x)^5). 3.2.5 Given y = [(3x)]. Step 1: Use the chain rule. Let u = (3x), so (du)/(dx) = -3(3x). Then y = u, so (dy)/(du) = (1)/(u). Step 2: Apply the chain rule. (dy)/(dx) = (1)/(u) · (du)/(dx) (dy)/(dx) = (1)/((3x)) · (-3(3x)) (dy)/(dx) = (-3(3x))/((3x)) Step 3: Simplify using trigonometric identities. (dy)/(dx) = -3(3x) The derivative is -3(3x). 3.2.6 Given y = -3e^5x + (2)/(5x^-2) - [3]x^5 + ()/(2). Step 1: Rewrite terms with exponents for differentiation. y = -3e^5x + (2)/(5)x^2 - x^5/3 + ()/(2) Step 2: Differentiate each term. Remember ()/(2) is a constant. (dy)/(dx) = (d)/(dx)(-3e^5x) + (d)/(dx)((2)/(5)x^2) - (d)/(dx)(x^5/3) + (d)/(dx)(()/(2)) (dy)/(dx) = -3(5e^5x) + (2)/(5)(2x) - (5)/(3)x^5/3 - 1 + 0 (dy)/(dx) = -15e^5x + (4)/(5)x - (5)/(3)x^2/3 Step 3: Express with positive exponents and in surd