Gradient and Hessian of $f(x,y) := a^T \left( x \odot \left[ \exp\left( \mu \ (y \ \oslash \ x) \right) - 1 \right] \right)$, wr.t. $x$ and $y$ How to find the Gradient and Hessian of
\begin{align}
f(x,y) := a^T \left( x \odot \left[ \exp\left( \mu \  (y \ \oslash \ x)  \right) - 1 \right] \right) \ ,
\end{align}
where $a, x, y \in \mathbb{R}^n$, all-ones vector $1 \in \mathbb{R}^n$, and $\mu  \in \mathbb{R}$? Also, $\odot$ and $\oslash$ means elementwise multiplication and division, respectively.
 A: Let
$$\eqalign{
z  &= y \oslash x\\
dz &= (dy \odot x - y \odot dx) \ \oslash \left( x \odot x \right) \\
}$$
and
$$\eqalign{
f &= a^T \left( x \odot \left[ \exp(\mu z) - 1\right] \right)\\
  &\equiv a : \left( x \odot \left[ \exp(\mu z) - 1\right] \right) \ ,
}$$
where for a scalar, trace function will output same scalar, then $\left\langle A, B \right\rangle={\rm tr}(A^TB) = A:B$.
Find the differential and then gradient:
$$\eqalign{
df &= \quad  a:   \left( dx \odot \left[ \exp(\mu z) - 1\right] \right) \\
    & \quad + \ a: \left( x \odot \left[ \mu \exp(\mu z) \odot \ dz \right] \right)\\
   &= \quad  a:   \left( dx \odot \left[ \exp(\mu z) - 1\right] \right) \\
    & \quad + \ a: \left( x \odot \left[ \mu \exp(\mu z) \odot \  (dy \odot x - y \odot dx) \ \oslash \left( x \odot x \right)  \right] \right)\\
}$$
To find $ \frac{\partial f}{\partial y}$, set $dx = 0$
$$\eqalign{
df &= a: \left( x \odot \left[ \mu \exp(\mu z) \odot \  (dy \odot x ) \ \oslash \left( x \odot x \right)  \right] \right)\\
   &=a : \exp\left( \mu \ y \oslash x \right) \odot   dy \\
   &=a \odot \exp\left( \mu \ y \oslash x \right) :    dy   
}$$
then,
$$\eqalign{
\frac{\partial f}{\partial y} &= a \odot \exp\left( \mu \ y \oslash x \right)  \ .
}$$
To find $ \frac{\partial f}{\partial x}$, set $dy = 0$
$$\eqalign{
df &= \quad  a:   \left( dx \odot \left[ \exp(\mu z) - 1\right] \right) \\
    & \quad + \ a: \left( x \odot \left[ \mu \exp(\mu z) \odot \  (- y \odot dx) \ \oslash \left( x \odot x \right)  \right] \right)\\
   &= \quad  a \odot \left( \exp(\mu \ y \oslash x) - 1\right) :   dx \\
   & \quad - \ \mu \ a \odot \left(y \oslash x  \right) \odot \exp(\mu \ y \oslash x): dx \ ,
}$$
then 
$$\eqalign{
\frac{\partial f}{\partial x} &= a \odot \left( \exp(\mu \ y \oslash x) - 1\right) - \mu \ a \odot \left(y \oslash x  \right) \odot \exp(\mu \ y \oslash x) \ .
}$$
