I was having problems with the question:
Find all first and second partial derivatives of f(x, t) = ln (x^2*t − t /x)?
Finding a partial derivative is just like finding an ordinary derivative. We differentiate with respect to the variable of the lower $\displaystyle \partial$ and treat all other independent variables as constants until we are done.
For example,
$\displaystyle \begin{aligned}
\frac{\partial f}{\partial x} &= \frac{\partial}{\partial x}\ln\left(x^2t-\frac{t}{x}\right)\\
&= \frac{1}{x^2t-\frac{t}{x}}\cdot\left(2xt+\frac{t}{x^2}\right)\;\ ;\;\;\;\;\;\;\;\;x^2t-\frac{t}{x}>0.
\end{aligned}
$
Hope this helps.