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: Rewrite the function in a form suitable for differentiation. The given function is y = (sqrt(x)-1)^3. We can rewrite sqrt(x) as x^1/2. So, y = (x^1/2-1)^3. Step 2: Apply the chain rule. Let u = x^1/2-1. Then y = u^3. The chain rule states that (dy)/(dx) = (dy)/(du) · (du)/(dx). First, find (dy)/(du): (dy)/(du) = (d)/(du)(u^3) = 3u^2 Next, find (du)/(dx): (du)/(dx) = (d)/(dx)(x^1/2-1) = (1)/(2)x^(1/2)-1 - 0 = (1)/(2)x^-1/2 Step 3: Substitute u back into the expression and multiply the derivatives. Substitute u = x^1/2-1 back into 3u^2: 3u^2 = 3(x^1/2-1)^2 Now, multiply (dy)/(du) and (du)/(dx): (dy)/(dx) = 3(x^1/2-1)^2 · (1)/(2)x^-1/2 Step 4: Simplify the expression. Rewrite x^1/2 as sqrt(x) and x^-1/2 as (1)/(sqrt(x)): (dy)/(dx) = 3(sqrt(x)-1)^2 · (1)/(2sqrt(x)) (dy)/(dx) = 3(sqrt(x)-1)^22sqrt(x) The final answer is 3(sqrt(x)-1)^22sqrt(x). That's 2 down. 3 left today — send the next one.