Differentiate normalization as a tangent projection

Define $N:\mathbb R^3\setminus\{\mathbf0\}\to\mathbb R^3$ by \[ N(\mathbf v)=\frac{\mathbf v}{\|\mathbf v\|}, \] and let $\mathbf p=(1,2,2)$. Derive a formula for the linear map $DN_{\mathbf p}(\mathbf h)$ and write its $3\times3$ matrix in the standard basis. Decompose an arbitrary increment into components parallel and perpendicular to $\mathbf p$, then prove that the derivative kills the radial component and scales the tangent component by $1/3$. Evaluate the derivative on the radial increment $\mathbf p$ and on the tangent increment $(2,-1,0)$. Finally, give the first-order approximation to $N(\mathbf p+\varepsilon(1,0,0))$ and verify directly that its first-order correction is tangent to the unit sphere at $N(\mathbf p)$.

Here $\|\mathbf p\|=3$ and $D(\mathbf v\cdot\mathbf v)_{\mathbf p}(\mathbf h)=2\mathbf p\cdot\mathbf h$.