  In previous work, the first author obtained conjecturally sharp upper bounds
for the joint moments of the $(2k-2h)^{\text{th}}$ power of the Riemann zeta
function with the $2h^{\text{th}}$ power of its derivative on the critical line
in the range $1\leq k \leq 2$, $0 \leq h \leq 1$. Unconditionally, we extend
these upper bounds to all $0 \leq h\leq k \leq 2$, and obtain lower bounds for
all $0\leq h \leq k+1/2$. Assuming the Riemann hypothesis, we give sharp bounds
for all $0\leq h \leq k$. We also prove upper bounds of the conjectured order
for more general joint moments of zeta with its higher derivatives.
