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
Welcome back — missed you this week. I will solve question 8. Question 8: Find the derivative of y = ((x (x))/(sqrt(1-x^2))). Step 1: Simplify the expression using logarithm properties. The property ((A)/(B)) = (A) - (B) and (AB) = (A) + (B) will be used. y = (x (x)) - (sqrt(1-x^2)) y = (x) + ((x)) - ((1-x^2)^1/2) y = (x) + ((x)) - (1)/(2)(1-x^2) Step 2: Differentiate each term with respect to x. Recall the differentiation rules: (d)/(dx)((u)) = (1)/(u) (du)/(dx) (d)/(dx)((x)) = (x) (d)/(dx)(x) = 1 (d)/(dx)(1-x^2) = -2x Differentiate (x): (d)/(dx)((x)) = (1)/(x) Differentiate ((x)): (d)/(dx)(((x))) = (1)/((x)) · (x) = (x) Differentiate -(1)/(2)(1-x^2): (d)/(dx)(-(1)/(2)(1-x^2)) = -(1)/(2) · (1)/(1-x^2) · (-2x) = (x)/(1-x^2) Step 3: Combine the derivatives of each term. (dy)/(dx) = (1)/(x) + (x) + (x)/(1-x^2) The final answer is (1)/(x) + (x) + (x)/(1-x^2). Send me the next one 📸