Use differentiation rules to determine the derivatives of each of the following, leaving the answer with positive exponent(s) and in surd form where applicable:

Here are the derivatives for each function: 3.2.1 Given the function: y = 4e^(x)/(2) + (1)/(2) x - (3)/(sqrt(x)) + 30^ Step 1: Rewrite terms with exponents for easier differentiation. y = 4e^(x)/(2) + (1)/(2) x - 3x^-(1)/(2) + 30^ Step 2: Differentiate each term using the appropriate rules ((d)/(dx)(e^ax) = ae^ax, (d)/(dx)( x) = (1)/(x), power rule, and derivative of a constant is 0). (dy)/(dx) = 4 ((1)/(2)e^(x)/(2)) + (1)/(2)((1)/(x)) - 3(-(1)/(2)x^-(3)/(2)) + 0 Step 3: Simplify the expression and write with positive exponents and surd form. (dy)/(dx) = 2e^(x)/(2) + (1)/(2x) + (3)/(2)x^-(3)/(2) (dy)/(dx) = 2e^(x)/(2) + (1)/(2x) + (3)/(2xsqrt(x)) The final answer is: (dy)/(dx) = 2e^(x)/(2) + (1)/(2x) + (3)/(2xsqrt(x)) 3.2.2 Given the function: y = (6x^4 - 3x^2)/(x^2) Step 1: Simplify the function by dividing each term in the numerator by x^2. y = (6x^4)/(x^2) - (3x^2)/(x^2) y = 6x^2 - 3 Step 2: Differentiate the simplified function using the power rule. (dy)/(dx) = 6(2x) - 0 (dy)/(dx) = 12x The final answer is: (dy)/(dx) = 12x 3.2.3 Given the function: y = 3x^3 (2x) Step 1: Use the product rule (d)/(dx)(uv) = u'v + uv'. Let u = 3x^3 and v = (2x). Step 2: Find the derivatives of u and v. u' = (d)/(dx)(3x^3) = 9x^2 v' = (d)/(dx)((2x)) = (2x) · (d)/(dx)(2x) = 2(2x) Step 3: Apply the product rule. (dy)/(dx) = (9x^2)((2x)) + (3x^3)(2(2x)) Step 4: Simplify the expression. (dy)/(dx) = 9x^2(2x) + 6x^3(2x) The final answer is: (dy)/(dx) = 9x^2(2x) + 6x^3(2x) 3.2.4 Given the function: y = (3)/((2-x^2)^4) Step 1: Rewrite the function using a negative exponent. y = 3(2-x^2)^-4 Step 2: Use the chain rule (d)/(dx)(f(g(x))) = f'(g(x)) · g'(x). Let f(u) = 3u^-4 and u = 2-x^2. f'(u) = 3(-4)u^-5 = -12u^-5 g'(x) = (d)/(dx)(2-x^2) = -2x Step 3: Apply the chain rule. (dy)/(dx) = (-12(2-x^2)^-5)(-2x) Step 4: Simplify the expression and write with positive exponents. (dy)/(dx) = 24x(2-x^2)^-5 (dy)/(dx) = (24x)/((2-x^2)^5) The final answer is: (dy)/(dx) = (24x)/((2-x^2)^5) 3.2.5 Given the function: y = ( 3x) Step 1: Use the chain rule for logarithmic functions (d)/(dx)((f(x))) = (f'(x))/(f(x)). Here f(x) = 3x. Step 2: Find the derivative of f(x). f'(x) = (d)/(dx)( 3x) = -(3x) · (d)/(dx)(3x) = -3(3x) Step 3: Apply the chain rule. (dy)/(dx) = (-3(3x))/((3x)) Step 4: Simplify using the identity = ( )/( ). (dy)/(dx) = -3(3x) The final answer is: (dy)/(dx) = -3(3x) 3.2.6 Given the function: y = -3e^5x + (2)/(5x^-2) - sqrt(x^5)2 + ( x)/(2) Step 1: Rewrite terms with exponents for easier differentiation. y = -3e^5x + (2)/(5)x^2 - (1)/(2)x^(5)/(2) + ()/(2)x Step 2: Differentiate each term using the appropriate rules ((d)/(dx)(e^ax) = ae^ax and power rule). (dy)/(dx) = -3(5e^5x) + (2)/(5)(2x) - (1)/(2)((5)/(2)x^(3)/(2)) + ()/(2)(1) Step 3: Simplify the expression and write with positive exponents and surd form. (dy)/(dx) = -15e^5x + (4)/(5)x - (5)/(4)x^(3)/(2) + ()/(2) (dy)/(dx) = -15e^5x + (4)/(5)x - (5)/(4)xsqrt(x) + ()/(2) The final answer is: (dy)/(dx) = -15e^5x + (4)/(5)x - (5)/(4)xsqrt(x) + ()/(2) Send me the next one 📸