Let $g$ a well defined function and $$f(u,v,w) = \int_u^{v}g(x,w)dx $$ I need to find $ \dfrac{\partial}{\partial u}f(u,v,w) $ and $ \dfrac{\partial}{\partial v}f(u,v,w)$. My approach was to use fundamental calculus theorem to conclude that $$ \dfrac{\partial}{\partial u}f(u,v,w) = \dfrac{\partial}{\partial u}\int_u^{v}g(x,w)dx = -\dfrac{\partial}{\partial u}\int_v^{u}g(x,w)dx = -g(u,w)$$ and $$ \dfrac{\partial}{\partial v}f(u,v,w) = \dfrac{\partial}{\partial u}\int_u^{v}g(x,w)dx = g(v,w)$$ Does the problem ends here? There's a remark about Leibniz rule, but i don't know how to use it in this case.
Thanks in advance