I'm working on a proof that requires me to simplify the derivative of a positive definite matrix. I'm relatively new to matrix calculus so I have been searching the internet for a good example. I haven't been able to find a good explanation though or step by step example of finding the derivative of a matrix.
For example, suppose I have a matrix
$$\left(\begin{array}{cc}a + x&c\\c&b + x\end{array}\right) $$
and I wanted to take the derivative of it with respect to x with scalars a,b,c. How would I go about this and where would I start?
Thanks in advance
Edit
intuitively,
$$ \left(\begin{array}{cc}a + x&c\\c&b + x\end{array}\right) = \left(\begin{array}{cc}a &c\\c&b\end{array}\right) + \left(\begin{array}{cc}x&0\\0&x\end{array}\right) $$ so $$ \begin{align} \frac{d}{dx} \left(\begin{array}{cc}a + x& c\\c&b + x\end{array}\right) &= \frac{d}{dx} \left(\begin{array}{cc}a&c\\c&b\end{array}\right) + \frac{d}{dx} \left(\begin{array}{cc}x&0\\0&x\end{array}\right) \\ &= 0 + \frac{d}{dx} \big(\begin{array}{cc}x\hat{\imath} + x\hat{\jmath}\end{array}\big) \\ &= \big(\begin{array}{cc}\hat{\imath} + \hat{\jmath}\end{array}\big) \\ &= \left(\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right) \end{align} $$
Is this correct?