Af = f(To + Ax) - f(IO). (5) Now we put together differentials and linear approximations. If f(x) is differentiable at c = a we have f'(a) = lim f(a + Ax) - f(x) : lim Af Ar-0 Ar-0 Ar So for Ar small we can write Af ~ f'(a)Ar. Date: March 3, 2021. 1 2 PROF. LAZAROS KIKAS DEPARTMENT OF MATHEMATICS UNIVERSITY OF DETROIT MERCY If we let dr = Ar, then Af ~ f'(a)Ax = f'(a)dx = dy. That is f(x + Ax) - f(x) ~ f'(a)Ax