  We prove gradient estimates for transition Markov semigroups $(P_t)$
associated to SDEs driven by multiplicative Brownian noise having possibly
unbounded $C^1$-coefficients, without requiring any monotonicity type
condition. In particular, first derivatives of coefficients can grow
polynomially and even exponentially. We establish pointwise estimates with
weights for $D_x P_t\varphi$ of the form \[ {\sqrt{t}} \, |D_x P_t \varphi (x)
| \le c \, (1+ |x|^k) \, \| \varphi\|_{\infty} \] $t \in (0,1]$, $\varphi \in
C_b ({\mathbb R}^d)$, $x \in {\mathbb R}^d.$ To prove the result we use two
main tools. First, we consider a Feynman--Kac semigroup with potential $V$
related to the growth of the coefficients and of their derivatives for which we
can use a Bismut-Elworthy-Li type formula. Second, we introduce a new regular
approximation for the coefficients of the SDE. At the end of the paper we
provide an example of SDE with additive noise and drift $b$ having sublinear
growth together with its derivative such that uniform estimates for $D_x P_t
\varphi$ without weights do not hold.
