Find \(f_x, f_y, f_{xx}, f_{yy}, f_{xy},\) and \(f_{yx}\) for \(f(x, y) = \ln{\left(x ^ 2 + y\right)}\).
All the examples (except \(f_y\)) will need to make use of the chain rule in that if \(h(\mathbf{\theta}) = f(g(\mathbf{\theta}))\) then \(h'(\mathbf{\theta}) = f'(g(\mathbf{\theta}))\cdot g'(\mathbf{\theta})\).
\[ \begin{aligned} \frac{\partial f}{\partial x} &= \frac{2x}{x ^ 2 + y}\\ \frac{\partial f}{\partial y} &= \frac{1}{x ^ 2 + y}\\ \frac{\partial^2 f}{\partial x^2} &= \frac{2(y-x^2)}{\left(x^2 + y\right)^2}\\ \frac{\partial^2 f}{\partial y^2} &= -\frac{1)}{\left(x^2 + y\right)^2}\\ \frac{\partial^2 f}{\partial x\partial y} &= -\frac{2x}{x ^ 2 + y} \end{aligned} \]
What were the most valuable elements you took away from this course?
I would list two most valuable elements which I took away from this course as of this point. The first was the review of linear algebra. In my career, I deal with probability and statistics daily and calculus often. I have had much need for, or done much work with, linear algebra over the past 25 years, so the review was valuable.
The second was the discipline required to not only post one’s own work weekly, but to comment on another student’s work as well. This required reviewing and thinking a wider array of problems than I would have chosen on my own, and thus learning new ideas, techniques, and approaches.