Fraction with components of Lorentz transformation

Question

I want to show how partial derivative transforms under a Lorentz transformation. Since the partial derivative has a fixed definition with respect to the $x$-coordinate it stays unchanged: $\partial_\mu\phi(x)\rightarrow\partial_\mu\phi(\Lambda x)$. With $x^{\prime}=\Lambda x$ we get then: $$\frac{\partial}{\partial x^\mu}=\frac{\partial}{\partial ((\Lambda^{-1})^\mu{}_\nu\Lambda^\nu{}_\rho x^\rho)}=\frac{\partial}{\partial((\Lambda^{-1})^\mu{}_{\nu}x^{\prime\nu})}= \Lambda_\mu{}^{\nu}\frac{\partial}{\partial x^{\prime\nu}}$$

How can one prove the following equation: $$\frac{\partial}{\partial((\Lambda^{-1})^\mu{}_{\nu}x^{\prime\nu})}= \Lambda_\mu{}^{\nu}\frac{\partial}{\partial x^{\prime\nu}}$$

Answer 1

Technically this might be considered a different derivation, but I guess it goes more directly to the point. Instead of rewriting the variable you are differentiating with respect to, you can just use a chain rule. Notice that $$\frac{\partial}{\partial x^\mu} = \frac{\partial {x}^{\nu'}}{\partial x^\mu}\frac{\partial}{\partial {x}^{\nu'}} = \Lambda^{\nu'}{}_\mu \frac{\partial}{\partial {x}^{\nu'}}.$$