The linear approximation to $f$ is exactly true if $f '$ is constant for that means that $f$ is linear. The inaccuracy of the linear approximation to $f$ at $x 0$ at argument $x$ arises from the changes to $f '$ between arguments $x 0$ and $x$ .
If $f '$ is differentiable in the interval between $x 0$ and $x$ we can get a better approximation to $f$ at $x$ by making a linear approximation to $f '$ and using it to estimate the change to $f$ in the interval.
In short if $f '$ is differentiable in that interval we can compute its derivative, called the second derivative of $f$ with respect to $x$ and written as $f " ( x )$ or as $d 2 f d x 2$ or sometimes as $f ( 2 ) ( x )$ and use it to improve the estimate of $f$ .