A derivation of the decomposition formula $\mathbf{R}″\left(p\right)\=\mathrm{\rho }\prime \mathbf{T}\+\mathrm{\kappa } {\mathrm{\rho }}^2\mathbf{N}$ follows from the Frenet formulas

$\mathbf{T}\left(p\right)\=\mathbf{R}\prime \/\mathrm{\ρ}$ and $\mathbf{N}\=\frac{1}{\mathrm{\κ}} \frac{d\mathbf{T}}{\mathrm{ds}}\=\frac{1}{\mathrm{\κ} \mathrm{\ρ}} \mathbf{T}\prime \left(p\right)$ 

if they are written as

$\mathbf{R}\prime \=\mathrm{\ρ} \mathbf{T}$ and $\frac{d\mathbf{T}}{\mathrm{ds}}\=k \mathbf{N}$ 

Differentiating, with respect to $p$, the first of these, gives

$\mathbf{R}″\left(p\right)\=\left(\mathrm{\ρ} \mathbf{T}\right)\prime \=\mathrm{\ρ}\prime \left(p\right)\mathbf{T}\+\mathrm{\ρ} \mathbf{T}\prime \left(p\right)$ 

By the chain rule, the second term on the right becomes

$\mathbf{T}\prime \left(p\right)\=\frac{d\mathbf{T}}{\mathrm{dp}}\=\frac{d\mathbf{T}}{\mathrm{ds}}\frac{\mathrm{ds}}{\mathrm{dp}}\=\frac{d\mathbf{T}}{\mathrm{ds}} \mathrm{\ρ}$ 

Hence, $\mathbf{R}″\left(p\right)$ becomes

$\mathbf{R}″\left(p\right)\=\mathrm{\ρ}\prime \left(p\right)\mathbf{T}\+\mathrm{\ρ} \frac{d\mathbf{T}}{\mathrm{ds}} \mathrm{\ρ}$ 

which, upon replacing $\frac{d\mathbf{T}}{\mathrm{ds}}$ with $\mathrm{\κ} \mathbf{N}$, and dropping the explicit display of the independent variable $p$ on the right, becomes the desired result $\mathbf{R}″\left(p\right)\=\mathrm{\rho }\prime \mathbf{T}\+\mathrm{\kappa } {\mathrm{\rho }}^2\mathbf{N}$.