Vertical alignment with multiple columns in math mode

Question

How do I get the $v$ to align correctly with the $dv$ above it

\begin{alignat*}{4}
  u &= \ln\left(x^4-1\right)   && \qquad && \dif v &= \dif x \\
  \dif u &= \frac{4x^3}{x^4-1} && \qquad && v &= x
\end{alignat*}