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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You're on a roll — To differentiate y = sqrt(x)(x^2+2)^3 using the product rule, we use the formula (dy)/(dx) = u'v + uv'. Step 1: Identify u and v. Let u = sqrt(x) = x^1/2. Let v = (x^2+2)^3. Step 2: Find the derivative of u, denoted as u'. u' = (d)/(dx)(x^1/2) = (1)/(2)x^1/2 - 1 = (1)/(2)x^-1/2 = (1)/(2sqrt(x)) Step 3: Find the derivative of v, denoted as v'. This requires the chain rule. Let w = x^2+2. Then v = w^3. Using the chain rule, (dv)/(dx) = (dv)/(dw) · (dw)/(dx). (dv)/(dw) = (d)/(dw)(w^3) = 3w^2 (dw)/(dx) = (d)/(dx)(x^2+2) = 2x So, v' = 3(x^2+2)^2 · (2x) = 6x(x^2+2)^2 Step 4: Apply the product rule formula (dy)/(dx) = u'v + uv'. (dy)/(dx) = ((1)/(2)x^-1/2)(x^2+2)^3 + (x^1/2)(6x(x^2+2)^2) Step 5: Simplify the expression. Rewrite x^-1/2 as (1)/(sqrt(x)) and x^1/2 as sqrt(x). (dy)/(dx) = ((x^2+2)^3)/(2sqrt(x)) + 6xsqrt(x)(x^2+2)^2 Factor out common terms, which are (x^2+2)^2 and (1)/(2sqrt(x)). (dy)/(dx) = ((x^2+2)^2)/(2sqrt(x)) [ (x^2+2) + 2sqrt(x) · 6xsqrt(x) ] Simplify the term 2sqrt(x) · 6xsqrt(x): 2sqrt(x) · 6xsqrt(x) = 12x (sqrt(x))^2 = 12x · x = 12x^2 Substitute this back into the expression: (dy)/(dx) = ((x^2+2)^2)/(2sqrt(x)) [ (x^2+2) + 12x^2 ] Combine terms inside the brackets: (dy)/(dx) = ((x^2+2)^2)/(2sqrt(x)) [ x^2+2+12x^2 ] (dy)/(dx) = ((x^2+2)^2 (13x^2+2))/(2sqrt(x)) The differentiated expression is: (dy)/(dx) = ((x^2+2)^2 (13x^2+2))/(2sqrt(x)) What's next?