Sakurai Modern Quantum Mechanics (Second Edition) Eq. 1.7.15

Homework Statement

Reading Sakurai I cannot see how he gets to the end of 1.7.15 as below:

Homework Equations

dx'|x'><x'-dx' |α>
= ∫dx'|x'>{<x' |α>-Δx'∂/∂x'<x' |α>}

The Attempt at a Solution

I tried a Taylor expansion but cannot see how this is derived.

blue_leaf77
Homework Helper
That's indeed a Taylor expansion, truncated up to the first order. The author wants to express the shifted position eigenket as in terms of infinitesimal change.

Thank you for your reply, it has helped me understand manipulation of bra-kets.