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
Welcome back Ambe — been a while! Let's pick up where you left off. Here are the derivatives of the given functions using first principles: 1) f(x) = x^3 Step 1: Write the definition of the derivative from first principles. f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 2: Substitute f(x) = x^3 and f(x+h) = (x+h)^3 into the formula. f'(x) = _h 0 ((x+h)^3 - x^3)/(h) Step 3: Expand (x+h)^3 and simplify the numerator. (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 f'(x) = _h 0 ((x^3 + 3x^2h + 3xh^2 + h^3) - x^3)/(h) f'(x) = _h 0 (3x^2h + 3xh^2 + h^3)/(h) Step 4: Factor out h from the numerator and cancel it with the denominator. f'(x) = _h 0 (h(3x^2 + 3xh + h^2))/(h) f'(x) = _h 0 (3x^2 + 3xh + h^2) Step 5: Apply the limit as h 0. f'(x) = 3x^2 + 3x(0) + (0)^2 f'(x) = 3x^2 2) f(x) = (1)/(x) Step 1: Write the definition of the derivative from first principles. f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 2: Substitute f(x) = (1)/(x) and f(x+h) = (1)/(x+h) into the formula. f'(x) = _h 0 (1)/(x+h) - (1)/(x)h Step 3: Combine the fractions in the numerator. f'(x) = _h 0 (x - (x+h))/(x(x+h))h f'(x) = _h 0 (-h)/(x(x+h))h Step 4: Simplify the complex fraction. f'(x) = _h 0 (-h)/(h · x(x+h)) f'(x) = _h 0 (-1)/(x(x+h)) Step 5: Apply the limit as h 0. f'(x) = (-1)/(x(x+0)) f'(x) = -(1)/(x^2) 3) f(x) = sqrt(x) Step 1: Write the definition of the derivative from first principles. f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 2: Substitute f(x) = sqrt(x) and f(x+h) = sqrt(x+h) into the formula. f'(x) = _h 0 sqrt(x+h) - sqrt(x)h Step 3: Multiply the numerator and denominator by the conjugate of the numerator, which is sqrt(x+h) + sqrt(x). f'(x) = _h 0 (sqrt(x+h) - sqrt(x))(sqrt(x+h) + sqrt(x))h(sqrt(x+h) + sqrt(x)) Step 4: Expand the numerator using the difference of squares formula (a-b)(a+b) = a^2 - b^2. f'(x) = _h 0 ((x+h) - x)/(h(sqrt(x+h) + x)) f'(x) = _h 0 (h)/(h(sqrt(x+h) + x)) Step 5: Cancel h from the numerator and denominator. f'(x) = _h 0 (1)/(sqrt(x+h) + x) Step 6: Apply the limit as h 0. f'(x) = (1)/(sqrt(x+0) + x) f'(x) = (1)/(sqrt(x) + x) f'(x) = (1)/(2sqrt(x)) Drop the next question!