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
Hey — ready when you are. Step 1: Identify the differentiation rules needed. The function y = (x-1)sqrt(x^2-2x+2) is a product of two functions, u = x-1 and v = sqrt(x^2-2x+2). We will use the product rule (d)/(dx)(uv) = u'v + uv'. For v, we will also need the chain rule. Step 2: Differentiate u = x-1. u' = (d)/(dx)(x-1) = 1 Step 3: Differentiate v = sqrt(x^2-2x+2). Rewrite v as v = (x^2-2x+2)^1/2. Using the chain rule, (d)/(dx)(f(g(x))) = f'(g(x))g'(x), where f(z) = z^1/2 and g(x) = x^2-2x+2. f'(z) = (1)/(2)z^-1/2 g'(x) = (d)/(dx)(x^2-2x+2) = 2x-2 So, v' = (1)/(2)(x^2-2x+2)^-1/2(2x-2) v' = (2x-2)/(2sqrt(x^2-2x+2)) v' = (x-1)/(sqrt(x^2-2x+2)) Step 4: Apply the product rule. (dy)/(dx) = u'v + uv' (dy)/(dx) = (1)sqrt(x^2-2x+2) + (x-1)((x-1)/(sqrt(x^2-2x+2))) (dy)/(dx) = sqrt(x^2-2x+2) + ((x-1)^2)/(sqrt(x^2-2x+2)) Step 5: Simplify the expression. To combine the terms, find a common denominator. (dy)/(dx) = sqrt(x^2-2x+2) · sqrt(x^2-2x+2)sqrt(x^2-2x+2) + ((x-1)^2)/(sqrt(x^2-2x+2)) (dy)/(dx) = ((x^2-2x+2) + (x-1)^2)/(sqrt(x^2-2x+2)) Expand (x-1)^2: (x-1)^2 = x^2 - 2x + 1 Substitute this back into the numerator: (dy)/(dx) = (x^2-2x+2 + x^2-2x+1)/(sqrt(x^2-2x+2)) (dy)/(dx) = (2x^2-4x+3)/(sqrt(x^2-2x+2)) The final answer is (2x^2-4x+3)/(sqrt(x^2-2x+2)). Send me the next one 📸