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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To find the derivative of y = x^2 - 5x + 4 from the first principle, we use the definition: (dy)/(dx) = _h 0 (f(x+h) - f(x))/(h) Step 1: Identify f(x) and find f(x+h). Given f(x) = x^2 - 5x + 4. Substitute x+h into f(x): f(x+h) = (x+h)^2 - 5(x+h) + 4 Expand the terms: f(x+h) = (x^2 + 2xh + h^2) - (5x + 5h) + 4 f(x+h) = x^2 + 2xh + h^2 - 5x - 5h + 4 Step 2: Calculate f(x+h) - f(x). f(x+h) - f(x) = (x^2 + 2xh + h^2 - 5x - 5h + 4) - (x^2 - 5x + 4) Distribute the negative sign: f(x+h) - f(x) = x^2 + 2xh + h^2 - 5x - 5h + 4 - x^2 + 5x - 4 Combine like terms: f(x+h) - f(x) = 2xh + h^2 - 5h Step 3: Divide by h. (f(x+h) - f(x))/(h) = (2xh + h^2 - 5h)/(h) Factor out h from the numerator: (f(x+h) - f(x))/(h) = (h(2x + h - 5))/(h) Cancel out h (since h ≠ 0 in the limit process): (f(x+h) - f(x))/(h) = 2x + h - 5 Step 4: Take the limit as h 0. (dy)/(dx) = _h 0 (2x + h - 5) Substitute h=0: (dy)/(dx) = 2x + 0 - 5 (dy)/(dx) = 2x - 5 The derivative of y = x^2 - 5x + 4 from the first principle is: 2x - 5 Send me the next one 📸