Suppose $A$ is a symmetric $n\times n$ matrix such that its eigenvalues are $\lambda_1,..., \lambda_n$. For $k=1,2,...,n$, define $\sigma_k(A)$ to be $$\sigma_k(A)=\sigma_k(\lambda_1,...,\lambda_n),$$ where $\sigma_k(\lambda_1,...,\lambda_n)$ is the $k$-th elementary symmetric function, i.e. $$\sigma_k(\lambda_1,...,\lambda_n)=\sum_{1\leq i_1<\cdots< i_k\leq n}\lambda_{i_1}\cdots\lambda_{i_k}.$$ For example, $$\sigma_1(A)=\lambda_1+\cdots+\lambda_n=tr(A)\mbox{ and } \sigma_n(A)=\lambda_1\lambda_2\cdots\lambda_n=\det(A).$$
My question is: Suppose the entries of $A$ depend on $t$ (so its eigenvalues also depend on $t$), I wonder if there is any formula for $\displaystyle\frac{d}{dt}\sigma_k(A)$.