I have the following formula and I would like to take the derivative of this function with respect to a where we know that $x$ is a vector of our data and $z$ is constant. how can I do this?
$$f(x) = \log\left(1 - {\left(e^{a\frac{x^2}{2}}\right)}^{e^{z}}\right)$$
You have the following:
$$ f(x, a) = \log\left(1 - e^{ae^z\frac{x^2}{2}}\right)\\ \frac{\partial f}{\partial a} = \frac{-e^z\frac{x^2}{2}e^{ae^z\frac{x^2}{2}}}{1 - e^{ae^z\frac{x^2}{2}}} = -\frac{x^2e^z}{2\left(e^{-\frac{ax^2e^z}{2}} - 1\right)} $$
