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 find the derivative of f(x) = x^2 sqrt(1 - x^2), we will use the product rule and the chain rule. Step 1: Identify the parts for the product rule. Let u(x) = x^2 and v(x) = sqrt(1 - x^2) = (1 - x^2)^1/2. The product rule states that if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x). Step 2: Find the derivative of u(x). u'(x) = (d)/(dx)(x^2) = 2x Step 3: Find the derivative of v(x) using the chain rule. v'(x) = (d)/(dx)((1 - x^2)^1/2) Apply the power rule and chain rule: v'(x) = (1)/(2)(1 - x^2)^-1/2 · (d)/(dx)(1 - x^2) v'(x) = (1)/(2)(1 - x^2)^-1/2 · (-2x) v'(x) = -x(1 - x^2)^-1/2 = (-x)/(sqrt(1 - x^2)) Step 4: Apply the product rule. f'(x) = u'(x)v(x) + u(x)v'(x) f'(x) = (2x)(sqrt(1 - x^2)) + (x^2)((-x)/(sqrt(1 - x^2))) f'(x) = 2xsqrt(1 - x^2) - (x^3)/(sqrt(1 - x^2)) Step 5: Combine the terms by finding a common denominator. The common denominator is sqrt(1 - x^2). f'(x) = 2xsqrt(1 - x^2) · sqrt(1 - x^2)sqrt(1 - x^2) - (x^3)/(sqrt(1 - x^2)) f'(x) = (2x(1 - x^2) - x^3)/(sqrt(1 - x^2)) f'(x) = (2x - 2x^3 - x^3)/(sqrt(1 - x^2)) f'(x) = (2x - 3x^3)/(sqrt(1 - x^2)) Step 6: Factor the numerator. f'(x) = (x(2 - 3x^2))/(sqrt(1 - x^2)) The final answer is (x(2 - 3x^2))/(sqrt(1 - x^2)). Send me the next one 📸