How to do partial derivatives with functions

Question

Let
$$U(x,t)=f(x+ct)$$ where $f$ is a differentiable function and $c\neq0$

How would I obtain $\frac{\partial U}{\partial t}$ and $\frac{\partial U}{\partial x}$?

Answer 1

Hint:

$$\dfrac{\partial f}{\partial t}=\dfrac{\partial f}{\partial (x+ct)}\dfrac{\partial (x+ct)}{\partial t}=c\cdot f'.$$

In which $f'$ denotes the derivative of $f$ with respect to $x+ct$.

Can you try the same method for the second derivative $\dfrac{\partial f}{\partial x}$?