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 differentiations for each function: vii) y = e^x^2+3 Step 1: Use the chain rule. Let u = x^2+3. Then (du)/(dx) = (d)/(dx)(x^2+3) = 2x. Step 2: Differentiate y = e^u with respect to u. (dy)/(du) = (d)/(du)(e^u) = e^u. Step 3: Apply the chain rule (dy)/(dx) = (dy)/(du) · (du)/(dx). (dy)/(dx) = e^x^2+3 · (2x) (dy)/(dx) = 2xe^x^2+3 viii) y = (x^2+2) Step 1: Use the chain rule. Let u = x^2+2. Then (du)/(dx) = (d)/(dx)(x^2+2) = 2x. Step 2: Differentiate y = u with respect to u. (dy)/(du) = (d)/(du)( u) = ^2 u. Step 3: Apply the chain rule (dy)/(dx) = (dy)/(du) · (du)/(dx). (dy)/(dx) = ^2(x^2+2) · (2x) (dy)/(dx) = 2x^2(x^2+2) ix) y = e^ x Step 1: Use the chain rule. Let u = x. Then (du)/(dx) = (d)/(dx)( x) = x. Step 2: Differentiate y = e^u with respect to u. (dy)/(du) = (d)/(du)(e^u) = e^u. Step 3: Apply the chain rule (dy)/(dx) = (dy)/(du) · (du)/(dx). (dy)/(dx) = e^ x · ( x) (dy)/(dx) = e^ x x x) y = (x) Step 1: Use the chain rule. Let u = sqrt(x) = x^1/2. Then (du)/(dx) = (d)/(dx)(x^1/2) = (1)/(2)x^-1/2 = (1)/(2sqrt(x)). Step 2: Differentiate y = u with respect to u. (dy)/(du) = (d)/(du)( u) = - u. Step 3: Apply the chain rule (dy)/(dx) = (dy)/(du) · (du)/(dx). (dy)/(dx) = -(sqrt(x)) · ((1)/(2sqrt(x))) (dy)/(dx) = -(x)2sqrt(x) xi) y = ^2(2sqrt(x)) Step 1: Rewrite the function as y = ((2sqrt(x)))^2. Step 2: Apply the chain rule. Let u = (2sqrt(x)). Then (dy)/(du) = (d)/(du)(u^2) = 2u = 2(2sqrt(x)). Step 3: Now find (du)/(dx) using the chain rule again. Let v = 2sqrt(x) = 2x^1/2. Then (dv)/(dx) = (d)/(dx)(2x^1/2) = 2 · (1)/(2)x^-1/2 = x^-1/2 = (1)/(sqrt(x)). Step 4: Differentiate u = v with respect to v. (du)/(dv) = (d)/(dv)( v) = - v = -(2sqrt(x)). Step 5: Apply the chain rule for (du)/(dx) = (du)/(dv) · (dv)/(dx). (du)/(dx) = -(2sqrt(x)) · ((1)/(sqrt(x))) = -(2sqrt(x))sqrt(x) Step 6: Apply the main chain rule (dy)/(dx) = (dy)/(du) · (du)/(dx). (dy)/(dx) = 2(2sqrt(x)) · (-(2sqrt(x))sqrt(x)) (dy)/(dx) = -2(2sqrt(x))(2sqrt(x))sqrt(x) Step 7: Use the trigonometric identity (2) = 2. Here = 2sqrt(x). So, 2(2sqrt(x))(2sqrt(x)) = (2 · 2sqrt(x)) = (4sqrt(x)). (dy)/(dx) = -(4sqrt(x))sqrt(x) xii) y = (()/(4) - 3x^2) Step 1: Use the chain rule. Let u = ()/(4) - 3x^2. Then (du)/(dx) = (d)/(dx)(()/(4) - 3x^2) = 0 - 3(2x) = -6x. Step 2: Differentiate y = u with respect to u. (dy)/(du) = (d)/(du)( u) = - u. Step 3: Apply the chain rule (dy)/(dx) = (dy)/(du) · (du)/(dx). (dy)/(dx) = -(()/(4) - 3x^2) · (-6x) (dy)/(dx) = 6x(()/(4) - 3x^2) xiii) y = ((x-3)/(x^2+3)) Step 1: Use the logarithm property ((A)/(B)) = A - B to simplify the expression. y = (x-3) - (x^2+3) Step 2: Differentiate each term with respect to x. For the first term, (d)/(dx)((x-3)): Let u = x-3, so (du)/(dx) = 1. (d)/(dx)( u) = (1)/(u) · (du)/(dx) = (1)/(x-3) · 1 = (1)/(x-3). For the second term, (d)/(dx)((x^2+3)): Let v = x^2+3, so (dv)/(dx) = 2x. (d)/(dx)( v) = (1)/(v) · (dv)/(dx) = (1)/(x^2+3) · 2x = (2x)/(x^2+3). Step 3: Combine the derivatives. (dy)/(dx) = (1)/(x-3) - (2x)/(x^2+3) Step 4: Combine into a single fraction. (dy)/(dx) = (1(x^2+3) - 2x(x-3))/((x-3)(x^2+3)) (dy)/(dx) = (x^2+3 - 2x^2+6x)/((x-3)(x^2+3)) (dy)/(dx) = (-x^2+6x+3)/((x-3)(x^2+3)) 3 done, 2 left today. You're making progress.