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.
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Here are the derivatives of the given functions using the first principle: 1) f(x) = x^2 - x Step 1: Write the definition of the derivative. f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 2: Substitute f(x+h) and f(x) into the formula. f'(x) = _h 0 (((x+h)^2 - (x+h)) - (x^2 - x))/(h) Step 3: Expand and simplify the numerator. f'(x) = _h 0 ((x^2 + 2xh + h^2 - x - h) - (x^2 - x))/(h) f'(x) = _h 0 (x^2 + 2xh + h^2 - x - h - x^2 + x)/(h) f'(x) = _h 0 (2xh + h^2 - h)/(h) Step 4: Factor out h from the numerator and cancel it with the denominator. f'(x) = _h 0 (h(2x + h - 1))/(h) f'(x) = _h 0 (2x + h - 1) Step 5: Evaluate the limit by substituting h=0. f'(x) = 2x + 0 - 1 = 2x - 1 The derivative is 2x-1. 2) f(x) = sqrt(x+1) Step 1: Write the definition of the derivative. f'(x) = _h 0 (f(x+h) - f(x))/(h) Step 2: Substitute f(x+h) and f(x) into the formula. f'(x) = _h 0 sqrt(x+h+1) - sqrt(x+1)h Step 3: Multiply the numerator and denominator by the conjugate of the numerator. f'(x) = _h 0 sqrt(x+h+1) - sqrt(x+1)h · sqrt(x+h+1) + sqrt(x+1)sqrt(x+h+1) + sqrt(x+1) Step 4: Simplify the numerator using (a-b)(a+b)=a^2-b^2. f'(x) = _h 0 ((x+h+1) - (x+1))/(h(sqrt(x+h+1) + x+1)) f'(x) = _h 0 (h)/(h(sqrt(x+h+1) + x+1)) Step 5: Cancel out h and evaluate the limit by substituting h=0. f'(x) = _h 0 (1)/(sqrt(x+h+1) + x+1) = (1)/(sqrt(x+0+1) + x+1) f'(x) = (1)/(sqrt(x+1) + x+1) = (1)/(2sqrt(x+1)) The derivative is (1)/(2sqrt(x+1)). 3) g(x) = (1)/(x^2) Step 1: Write the definition of the derivative. g'(x) = _h 0 (g(x+h) - g(x))/(h) Step 2: Substitute g(x+h) and g(x) into the formula. g'(x) = _h 0 (1)/((x+h)^2) - (1)/(x^2)h Step 3: Combine the fractions in the numerator. g'(x) = _h 0 (x^2 - (x+h)^2)/(x^2(x+h)^2)h g'(x) = _h 0 (x^2 - (x^2 + 2xh + h^2))/(h x^2(x+h)^2) g'(x) = _h 0 (-2xh - h^2)/(h x^2(x+h)^2) Step 4: Factor out h from the numerator and cancel it with the denominator. g'(x) = _h 0 (h(-2x - h))/(h x^2(x+h)^2) g'(x) = _h 0 (-2x - h)/(x^2(x+h)^2) Step 5: Evaluate the limit by substituting h=0. g'(x) = (-2x - 0)/(x^2(x+0)^2) = (-2x)/(x^2 · x^2) = (-2x)/(x^4) g'(x) = -(2)/(x^3) The derivative is -(2)/(x^3). 4) h(x) = (1)/(sqrt(x)) Step 1: Write the definition of the derivative. h'(x) = _h 0 (h(x+h) - h(x))/(h) Step 2: Substitute h(x+h) and h(x) into the formula. h'(x) = _h 0 (1)/(sqrt(x+h)) - (1)/(sqrt(x))h Step 3: Combine the fractions in the numerator. h'(x) = _h 0 sqrt(x) - sqrt(x+h)sqrt(x)sqrt(x+h)h = _h 0 sqrt(x) - sqrt(x+h)hsqrt(x)sqrt(x+h) Step 4: Multiply the numerator and denominator by the conjugate of the numerator. h'(x) = _h 0 sqrt(x) - sqrt(x+h)hsqrt(x)sqrt(x+h) · sqrt(x) + sqrt(x+h)sqrt(x) + sqrt(x+h) h'(x) = _h 0 (x - (x+h))/(hsqrt(x)x+h)(sqrt(x) + sqrt(x+h)) h'(x) = _h 0 (-h)/(hsqrt(x)x+h)(sqrt(x) + sqrt(x+h)) Step 5: Cancel out h and evaluate the limit by substituting h=0. h'(x) = _h 0 (-1)/(sqrt(x)x+h)(sqrt(x) + sqrt(x+h)) h'(x) = (-1)/(sqrt(x)x+0)(sqrt(x) + sqrt(x+0)) = (-1)/(sqrt(x)x)(2sqrt(x)) h'(x) = (-1)/(x(2sqrt(x))) = -(1)/(2xsqrt(x)) The derivative is -(1)/(2xsqrt(x)). 5) y = (1)/(2x-1) Step 1: Write the definition of the derivative. (dy)/(dx) = _h 0 (f(x+h) - f(x))/(h) Step 2: Substitute f(x+h) and f(x) into the formula. (dy)/(dx) = _h 0 (1)/(2(x+h)-1) - (1)/(2x-1)h (dy)/(dx) = _h 0 (1)/(2x+2h-1) - (1)/(2x-1)h Step 3: Combine the fractions in the numerator. (dy)/(dx) = _h 0 ((2x-1) - (2x+2h-1))/((2x+2h-1)(2x-1))h (dy)/(dx) = _h 0 (2x-1-2x-2h+1)/(h(2x+2h-1)(2x-1)) (dy)/(dx) = _h 0 (-2h)/(h(2x+2h-1)(2x-1)) Step 4: Cancel out h and evaluate the limit by substituting h=0. (dy)/(dx) = _h 0 (-2)/((2x+2h-1)(2x-1)) (dy)/(dx) = (-2)/((2x+2(0)-1)(2x-1)) = (-2)/((2x-1)(2x-1)) (dy)/(dx) = -(2)/((2x-1)^2) The derivative is -(2)/((2x-1)^2). That's 2 down. 3 left today — send the next one.