This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
ScanSolve AI Answer
Step 1: Identify the layers of the function for the chain rule. The function is y = ^2(sqrt((x))). This can be written as y = ((sqrt((x))))^2. Let u = (sqrt((x))). Then y = u^2. Let v = sqrt((x)). Then u = (v). Let w = (x). Then v = sqrt(w) = w^1/2. Step 2: Differentiate each layer. Differentiate y = u^2 with respect to u: (dy)/(du) = 2u Differentiate u = (v) with respect to v: (du)/(dv) = (v)(v) Differentiate v = w^1/2 with respect to w: (dv)/(dw) = (1)/(2)w^-1/2 = (1)/(2sqrt(w)) Differentiate w = (x) with respect to x: (dw)/(dx) = (x) Step 3: Apply the chain rule (dy)/(dx) = (dy)/(du) · (du)/(dv) · (dv)/(dw) · (dw)/(dx). Substitute back the expressions for u, v, w: u = (sqrt((x))) v = sqrt((x)) w = (x) (dy)/(dx) = 2((sqrt((x)))) · ((sqrt((x)))(sqrt((x)))) · ((1)/(2sqrt((x)))) · ((x)) Step 4: Simplify the expression. (dy)/(dx) = 2 ^2(sqrt((x))) (sqrt((x))) · ((x))/(2sqrt((x))) (dy)/(dx) = ^2(sqrt((x))) (sqrt((x))) (x)sqrt((x)) The final answer is ^2(sqrt((x))) (sqrt((x))) (x)sqrt((x)). That's 2 down. 3 left today — send the next one.