if i have a function
$\displaystyle c(x) = 5x + t $
The derivative with respect to x is 5 ...right?
That would be correct...it $\displaystyle t$ was a constant.
If $\displaystyle t$ is a function of x, then $\displaystyle c(x)=5x+t\implies c^{\prime}(x)=5+\frac{\,dt}{\,dx}$
If $\displaystyle t$ isn't a constant, but rather a different variable, then define $\displaystyle c(x,t)=5x+t$. Thus, the [partial] derivative with respect to x would be $\displaystyle c_x(x,t)=5$.**
Does this help?
** If you don't understand this part, that's ok. I'm just trying to cover all possibilities. Pick the one that best answers your question based off of where you're at with respect to differentiation.