matrix derivative w.r.t a scalar

Question

How the following derivative can be calculated?

$\displaystyle\frac{d}{d\lambda}A\left(\lambda\ I+A^TA\right)^{-1}A^T$

where $A$ is a rectangular matrix and $\lambda$ is a scalar.

Answer 1

$f'(\lambda)=-A(\lambda I+A^TA)^{-2}A^T$.