The first partial derivative of $w$ with respect to $x$ is:
$$w_x = \frac{\partial w}{\partial x} = \frac{1}{2x+2ct} \cdot 2 = \frac{1}{x+ct}.$$
The second partial derivative of $w$ with respect to $x$ is:
$$w_{xx} = \frac{\partial^2 w}{\partial x^2} =
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