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
You got it, Minnie🔥♥! Let's tackle 2.1. 2.1. Determine the derivative of f(x) by using first principles. Given f(x) = (2x^3)/(x^4). Step 1: Simplify f(x). f(x) = (2x^3)/(x^4) = 2x^3-4 = 2x^-1 = (2)/(x) Step 2: Apply the definition of the derivative from first principles. The derivative f'(x) is defined as: f'(x) = _h 0 (f(x+h) - f(x))/(h) Substitute f(x) = (2)/(x) and f(x+h) = (2)/(x+h) into the formula. f'(x) = _h 0 (2)/(x+h) - (2)/(x)h Step 3: Simplify the numerator by finding a common denominator. f'(x) = _h 0 (2x - 2(x+h))/(x(x+h))h f'(x) = _h 0 (2x - 2x - 2h)/(x(x+h))h f'(x) = _h 0 (-2h)/(x(x+h))h Step 4: Simplify the complex fraction and evaluate the limit. f'(x) = _h 0 (-2h)/(h · x(x+h)) Cancel out h from the numerator and denominator. f'(x) = _h 0 (-2)/(x(x+h)) Now, substitute h=0 into the expression. f'(x) = (-2)/(x(x+0)) f'(x) = (-2)/(x^2) The derivative of f(x) is: f'(x) = -(2)/(x^2) That's 2 down. 3 left today — send the next one.