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 are the derivatives for the given functions: 1) For f(x) = 2x · x Step 1: Identify the product rule. The product rule states that if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x). Let u(x) = 2x and v(x) = x. Step 2: Find the derivatives of u(x) and v(x). u'(x) = (d)/(dx)(2x) = 2 v'(x) = (d)/(dx)( x) = (1)/(x) Step 3: Apply the product rule. f'(x) = (2)( x) + (2x)((1)/(x)) Step 4: Simplify the expression. f'(x) = 2 x + 2 f'(x) = 2 x + 2 2) For g(x) = (5 - 2 (x-1))/(x) Step 1: Identify the quotient rule. The quotient rule states that if g(x) = (u(x))/(v(x)), then g'(x) = (u'(x)v(x) - u(x)v'(x))/([v(x)]^2). Let u(x) = 5 - 2 (x-1) and v(x) = x. Step 2: Find the derivative of u(x). u'(x) = (d)/(dx)(5 - 2 (x-1)) Using the chain rule for (x-1), where the derivative of (w) is (1)/(w) and the derivative of x-1 is 1: u'(x) = 0 - 2 · (1)/(x-1) · 1 = -(2)/(x-1) Step 3: Find the derivative of v(x). v'(x) = (d)/(dx)(x) = 1 Step 4: Apply the quotient rule. g'(x) = ((-2)/(x-1))(x) - (5 - 2 (x-1))(1)x^2 Step 5: Simplify the numerator. g'(x) = (-2x)/(x-1) - 5 + 2 (x-1)x^2 Step 6: Combine terms in the numerator by finding a common denominator. g'(x) = (-2x)/(x-1) - (5(x-1))/(x-1) + 2 (x-1)x^2 g'(x) = (-2x - 5x + 5)/(x-1) + 2 (x-1)x^2 g'(x) = (-7x + 5)/(x-1) + 2 (x-1)x^2 To express the numerator as a single fraction: g'(x) = (-7x + 5 + 2(x-1)(x-1))/(x-1)x^2 g'(x) = (-7x + 5 + 2(x-1)(x-1))/(x^2(x-1)) g'(x) = (-7x + 5 + 2(x-1)(x-1))/(x^2(x-1)) Send me the next one 📸