Given y = sqrt(x2 - 5x + 8).

You're on a roll — Here are the solutions for the circled problems: Problem 1.5: Given y = sqrt(x^2 - 5x + 8). Step 1: Rewrite the function using exponent notation. y = (x^2 - 5x + 8)^1/2 Step 2: Apply the chain rule. Let u = x^2 - 5x + 8. Then y = u^1/2. The derivative of y with respect to u is (dy)/(du) = (1)/(2)u^-1/2. The derivative of u with respect to x is (du)/(dx) = 2x - 5. Using the chain rule, (dy)/(dx) = (dy)/(du) · (du)/(dx). (dy)/(dx) = (1)/(2)(x^2 - 5x + 8)^-1/2 · (2x - 5) Step 3: Simplify the expression. (dy)/(dx) = (2x - 5)/(2sqrt(x^2 - 5x + 8)) The derivative is (2x - 5)/(2sqrt(x^2 - 5x + 8)). Problem 1.6: Given y = (1)/(sqrt(x^3 + 1)). Step 1: Rewrite the function using exponent notation. y = (x^3 + 1)^-1/2 Step 2: Apply the chain rule. Let u = x^3 + 1. Then y = u^-1/2. The derivative of y with respect to u is (dy)/(du) = -(1)/(2)u^-3/2. The derivative of u with respect to x is (du)/(dx) = 3x^2. Using the chain rule, (dy)/(dx) = (dy)/(du) · (du)/(dx). (dy)/(dx) = -(1)/(2)(x^3 + 1)^-3/2 · (3x^2) Step 3: Simplify the expression. (dy)/(dx) = -(3x^2)/(2(x^3 + 1)^3/2) (dy)/(dx) = -(3x^2)/(2sqrt((x^3 + 1)^3)) The derivative is -(3x^2)/(2sqrt((x^3 + 1)^3)). Problem 1.14: Given f(x) = ( 3x). Step 1: Apply the chain rule for the natural logarithm. The derivative of (g(x)) is (g'(x))/(g(x)). Here, g(x) = 3x. So, f'(x) = (1)/( 3x) · (d)/(dx)( 3x). Step 2: Apply the chain rule again for 3x. Let h(x) = 3x. Then 3x = (h(x)). The derivative of (h(x)) is (h(x)) · h'(x). So, (d)/(dx)( 3x) = (3x) · (d)/(dx)(3x). (d)/(dx)(3x) = 3. Therefore, (d)/(dx)( 3x) = 3 3x. Step 3: Substitute the derivative back into the expression for f'(x). f'(x) = (1)/( 3x) · (3 3x) f'(x) = (3 3x)/( 3x) Step 4: Simplify using trigonometric identities. Since ( )/( ) = : f'(x) = 3 3x The derivative is 3 3x. What's next?