  We shall develop a general technique to obtain the super heat kernel
coefficients of an arbitrary second order operator in $N=1$ superspace. We
focus on the space of conformal supergravity here but the method presented is
equally applicable for other types of superspace. The first three coefficients
which determine the one-loop divergence of the corresponding quantum theory
will be calculated. As an application we shall present the one-loop logarithmic
divergence of super Yang-Mills theory coupled to a string dilaton $S$. This is
the first superfield calculation for SYM with a non-trivial gauge kinetic
function, which generalize the previous result with a constant coupling
strength. We also demonstrate that the method presented can be extended to the
case of third order operators, with the restriction that its third order part
is composed of only spinor derivatives.
