$f : \mathbb{R}^2 \rightarrow \mathbb{R}, f(x, y) = \int_{0}^{y} \cos(t^2x) \, dt.$
I am facing difficulties in understanding the approach to solve this exercise.
I have tried reviewing my course materials and textbooks, but I am still unsure about the steps involved in analyzing partial differentiability and calculating the Jacobian matrix. I would greatly appreciate it if someone could provide guidance or insights into how to approach and solve this type of problem.
You need to use the Leibniz integral rule, allowed by the theorem of differentiability under the integral sign. It states under good conditions of the function that $$\frac{d}{dx}\left(\int_{a(x)}^{b(x)}f(x,t)dt\right)=f(x,b(x))\cdot b'(x)-f(x,a(x))\cdot a'(x)+\int_{a(x)}^{b(x)}\frac{\partial}{\partial t}f(x,t)dt.$$ That's for the variable $x$, for the $y$ just apply the FTC. So $$f_y=\cos(xy^2),\,f_{yy}=-2xy\sin(xy^2),$$ and for $x$ we get $$f_x=\int_{0}^y -t^2\sin(t^2 x)dt,\,f_{xx}=\int_{0}^y -t^4\cos (t^2x)dx.$$ Now the crossed partials $$f_{yx}=-y^2\sin (xy^2),\, f_{xy}=-y^2\sin (y^2x),$$ which are equal, a good indicator that we did the things well.