  We develop a new method, which is based on the optimal redshift weighting
scheme, to extract the maximal tomographic information of baryonic acoustic
oscillations (BAO) and redshift space distortions (RSD) from the extended
Baryon Oscillation Spectroscopic Survey (eBOSS) Data Release 14 quasar (DR14Q)
survey. We validate our method using the EZ mocks, and apply our pipeline to
the eBOSS DR14Q sample in the redshift range of $0.8<z<2.2$. We report a joint
measurement of $f\sigma_8$ and two-dimensional BAO parameters $D_{\rm A}$ and
$H$ at four effective redshifts of $z_{\rm eff}=0.98, 1.23, 1.52$ and $1.94$,
and provide the full data covariance matrix. Using our measurement combined
with BOSS DR12, MGS and 6dFGS BAO measurements, we find that the existence of
dark energy is supported by observations at a $7.4\sigma$ significance level.
Combining our measurement with BOSS DR12 and Planck observations, we constrain
the gravitational growth index to be $\gamma=0.580\pm0.082$, which is fully
consistent with the prediction of general relativity. This paper is part of a
set that analyses the eBOSS DR14 quasar sample.
