given y = x^2 - 7x + 5, find dy/dx from first principles
using the delta method for first principles derivation:
Define the differential: dy/dx = limit as h -> 0 f(x+h) - f(x)/h where f(x) = ysubstitute the equation into the differential: dy/dx = (x+h)^2 - 7*(x+h) + 5 - (x^2 - 7x +5)/hexpand the brackets to form quadratic: dy/dx = x^2 + h^2 + 2xh - 7x -7h + 5 - x^2 + 7x - 5/hcancel out the variables: dy/dx = h^2 + 2xh - 7h / hdivide by h: dy/dx = h + 2x - 7Finish it off by taking the limit of h to be 0: dy/dx = 2x - 7 Simple method to follow with an example for a 5 mark question that consistently comes up in core 1.
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