Definition of the derivative:
df(x)/dx = lim h→0 [ (f(x+h)-f(x))/h ]
f(x) = -3x2 + 7x + 5
f(x+h) = -3(x+h)2 + 7(x+h) + 5 = -3(x2+2xh+h2) + 7x + 7h + 5 = -3x2 - 6xh - 3h2 + 7x + 7h + 5
Plugging f(x+h) and f(x) into the derivative definition:
d f(x)/dx = lim h→0 (-3x2 - 6xh - 3h2 + 7x + 7h + 5 + 3x2 - 7x - 5)/h
= lim h→0 (-6xh - 3h2 + 7h)/h
= lim h→0 (-6x - 3h + 7)
Now lim h→0 (-3h) = 0, so:
d f(x)/dx = -6x + 7
