Lorentz transforming differential operators on scalar fields

[Theage]

Homework Statement

I'm reading Peskin and Schroeder to the best of my ability. Other than a few integration tricks that escaped me I made it through chapter 2 with no trouble, but the beginning of chapter three, "Lorentz Invariance in Wave Equations", has me stumped. They are going through a proof that the Klein-Gordon equation (\partial_\mu\partial^\mu+m^2)\phi=0 is Lorentz invariant, and I can't understand for the life of me how they came up with the transformations of differential operators.

Homework Equations

\partial_\mu\phi(x)\to\partial_\mu(\phi(\Lambda^{-1}x))=(\Lambda^{-1})^\nu_{\,\,\mu}(\partial_\nu\phi)(\Lambda^{-1}x)

The Attempt at a Solution

I understand (at least intuitively) how a scalar field should transform on its own, which makes it easy to transform the mass term in the KG Lagrangian (we're working with no potential or interaction terms) which is proportional to the square of the field. The second equality in the equation above is the problem. At first I thought it had something to do with conjugation by the inverse of the transformation in question, but this doesn't seem right.

 

[Orodruin]

Try the chain rule $\partial_\mu = (\partial x'{}^\nu/\partial x^\mu) \partial_{\nu'}$ and the relation between $\partial^\mu$ and $\partial_\mu$.