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
Here's the solution to the differentiation problem: We are given the function f(x) = x^2sqrt(1-x^2) and need to show that f'(x) = (x(2-3x^2))/(sqrt(1-x^2)). Step 1: Identify the components for the product rule. Let u = x^2 and v = sqrt(1-x^2) = (1-x^2)^1/2. Step 2: Differentiate u and v. Differentiating u: (du)/(dx) = 2x Differentiating v using the chain rule: (dv)/(dx) = (1)/(2)(1-x^2)^-1/2 · (-2x) (dv)/(dx) = -x(1-x^2)^-1/2 (dv)/(dx) = (-x)/(sqrt(1-x^2)) Step 3: Apply the product rule, f'(x) = u(dv)/(dx) + v(du)/(dx). f'(x) = (x^2)((-x)/(sqrt(1-x^2))) + (sqrt(1-x^2))(2x) f'(x) = (-x^3)/(sqrt(1-x^2)) + 2xsqrt(1-x^2) Step 4: Combine the terms by finding a common denominator. The common denominator is sqrt(1-x^2). f'(x) = (-x^3)/(sqrt(1-x^2)) + 2xsqrt(1-x^2) · sqrt(1-x^2)sqrt(1-x^2) f'(x) = (-x^3 + 2x(1-x^2))/(sqrt(1-x^2)) Step 5: Simplify the numerator. f'(x) = (-x^3 + 2x - 2x^3)/(sqrt(1-x^2)) f'(x) = (2x - 3x^3)/(sqrt(1-x^2)) Step 6: Factor out x from the numerator. f'(x) = (x(2 - 3x^2))/(sqrt(1-x^2)) This matches the expression we needed to show. 3 done, 2 left today. You're making progress.