The strategy is to write the expression as a scalar using index notation, take the
derivative, and re-write in matrix form.

Note that to write the function as a summation of matrices we have to write just one scalar as a matrix multiplication because the function is scalar:
$$
f(M)= [x^TMx]_{11}= \sum_i x_{i1}[Mx]_{i1}
$$
At first summation, we have $x_{i1}$ because we have $x^T$ as the first term.

Now expand $[Mx]_{i1}$
$$
f(M)= [x^TMx]_{11}= \sum_i x_{i1}\sum_j M_{ij}x_{j1}=\sum_i \sum_j x_{i1} M_{ij}x_{j1}
$$

Now take the derivative with respect to $M_{ij}$

$$
\frac{\partial f(M)}{\partial M_{ij}}=\sum_i \sum_j x_{i1}x_{j1}
$$

Looking at the indices, we can see that

$$
\sum_i \sum_j x_{i1}x_{j1}=\sum_i \sum_j x_{j1}x_{i1}=[xx^T]_{ji}
$$

Therefore,

$$
\frac{\partial f(M)}{\partial M_{ij}}=xx^T
$$