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We study the slopes of the upper critical field$\partial_{T}H_{c2}|_{T_{c}}\equiv\partial H_{_{c2}}/\partial T$ at $T_{c}$ inanisotropic superconductors with transport (non-magnetic) scattering employingthe Ginzburg-Landau theory, developed for this situation by S. Pokrovsky and V.Pokrovsky, Phys. Rev. B 54, 13275 (1996). We found unexpected behavior of theslopes for a $d-$wave superconductor and in a more general case of materialswith line nodes in the order parameter. Specifically, the presence of linenodes causes $\partial_{T}H_{c2}|_{T_{c}}$ to decrease with increasingnon-magnetic scattering parameter $P$, unlike the nodeless case where the slopeincreases. In a pure $d-$wave case, the slope $\partial H_{c2}|_{T_{c}}$changes from decreasing to increasing when scattering parameter approaches$P\approx0.91\,P_{{\rm crit}}$, where $P_{{\rm crit}}\approx0.2807$ at which$T_{c}\to0$ that implies the the existence of a gapless state in $d-$wavesuperconductors with transport scattering in the interval, $0.91\,P_{{\rm{crit}}}<P<P_{{\rm crit}}$. Furthermore, we have considered the mixed $s+d$order parameter that has 4 nodes on a cylindrical Fermi surface when a $d-$partis dominant, or no nodes at all when an $s-$phase is the major one. We findthat presence of nodes causes the slope $\partial_{T}H_{c2}|_{T_{c}},$ todecrease initially with increasing $P$, whereas in the nodeless state,$\partial_{T}H_{c2}|_{T_{c}}$ monotonically increases. Therefore, fairlystraightforward experiments make it possible to decide whether or not the orderparameter of a superconductor has nodes by measuring the disorder-dependence ofthe slope of $H_{c2}$ at $T_{c}$.

Disorder-dependent slopes of the upper critical field in nodal and nodeless superconductors