Let
f(x)={−2x2+7x, for x<0
8x2−5, for x⩾0.
According to the definition of the derivative, to compute f′(0), we need to compute the left-hand limit
lim=
x→0−
lim=
x→0+
f(x)=
{−2x^2+7x, for x<0
{8x^2−5, for x⩾0
f'(x) =
{-4x+7 for x<0
{16x for x⩾0
lim(x→0−) = 7
lim(x→0+) = 0
so f'(0) does not exist