I'm studying some topics on PDE and I'm having trouble with a really basic thing... how embarrassing!
I'm trying to compute the following derivative: $$\frac{d}{dt}u(t,x)$$ where $$u(t,x)=\phi(x-ct)+\int_0^tf(s,x-ct+cs)ds$$ My answer was $$\frac{d}{dt}u(t,x)=-c\phi'(x+ct)+f(t,x)+\int_0^t\frac{\partial}{\partial t}f(s,x-ct+cs)ds$$ However, my professor wrote:$$\frac{d}{dt}u(t,x)=-c\phi'(x+ct)+f(t,x)+-c\int_0^t\frac{\partial}{\partial x}f(s,x-ct+cs)ds$$ I was wondering: does the following hold? $$\frac{\partial}{\partial t}f(s,x-ct+cs)=-c\frac{\partial}{\partial x}f(s,x-ct+cs)$$ My guess was that $$ \frac{\partial}{\partial t}f(s,g(x,t,s))=\frac{\partial}{\partial s}f(s,g(x,t,s))\cdot\frac{\partial}{\partial t}(s)+\frac{\partial f}{\partial g}\frac{\partial g}{\partial t} = 0+(-c)\frac{\partial f}{\partial g} $$ but I don't see how this could help... can someone help me, please?