I'm having trouble figuring out where my error is for the following computation.
Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$, and define the measure $\mu$ via $\mu(A) = \lambda(A^3)$. Similarly define $\nu(A) = \mu(A^3) = \lambda(A^9)$. Note all three measures are equivalent so $\lambda$-a.e., $\mu$-a.e., $\nu$-a.e. all coincide. Then on one hand
$$ \nu([a,b]) = \lambda([a^9, b^9]) = b^9 - a^9 = \int_{a^9}^{b^9} 9x^8 \, d\lambda $$
Therefore, $\frac{d\nu}{d\lambda} = 9x^8$ ($\lambda$-a.e.) where we use the Radon-Nikodym derivative; $x$ is a dummy variable. By essentially the same computation, $\frac{d\nu}{d\mu} = \frac{d\mu}{d\lambda} = 3x^2$ ($\lambda$-a.e.). We also have the chain rule
$$ \frac{d\nu}{d\lambda} = \frac{d\nu}{d\mu} \frac{d\mu}{d\lambda}. \tag{$\ast$} $$
But as computed, the LHS is $9x^8$ and the RHS is $9x^4$. What am I doing wrong? Conceptually, it seems like the chain rule should give $9x^8 = 3(x^3)^2 \cdot 3x^2$ where we've composed the first term with $x \mapsto x^3$. But in ($\ast$) we just have a plain product of functions.