Derivative of a matrix inverse with respect to a parameter Given some real symmetric positive semidefinite matrix $X\in\mathbb{R}^{n\times n}$ and a parameter $\lambda \in\mathbb{R}_+$, I want to differentiate:
$$f(\lambda) = (\lambda \mathrm{Id} + X)^{-1}$$
but I don't know where to start. Clearly $f:\mathbb{R}_+\to \mathbb{R}^{n\times n}$ but not sure how to handle the inverse.

Let $\lambda \in\mathbb{R}_+$ and $ \lambda I+X$ be invertible. Then there is an open neighborhood $U$ of $ \lambda$ such that $ \mu I+X$ is invertible for all $ \mu \in U.$
Now show that for $ \mu \in U$ with $ \lambda \ne \mu$ we have
$$\frac{f(\lambda)-f(\mu)}{\lambda - \mu}= -f(\lambda)f( \mu).$$
Can you proceed ?
