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.
ScanSolve AI Answer
To differentiate y = (sqrt(x)-1)^3 with respect to x, we will use the chain rule. Step 1: Let u = sqrt(x)-1. Then y = u^3. Step 2: Differentiate u with respect to x. We can write sqrt(x) as x^1/2. (du)/(dx) = (d)/(dx)(x^1/2-1) = (1)/(2)x^(1/2)-1 - 0 = (1)/(2)x^-1/2 = (1)/(2sqrt(x)) Step 3: Differentiate y with respect to u. (dy)/(du) = (d)/(du)(u^3) = 3u^2 Step 4: Apply the chain rule, which states (dy)/(dx) = (dy)/(du) · (du)/(dx). (dy)/(dx) = (3u^2) · ((1)/(2sqrt(x))) Step 5: Substitute u = sqrt(x)-1 back into the expression. (dy)/(dx) = 3(sqrt(x)-1)^2 · (1)/(2sqrt(x)) Step 6: Simplify the expression. (dy)/(dx) = 3(sqrt(x)-1)^22sqrt(x) The derivative is: 3(sqrt(x)-1)^22sqrt(x) What's next?