Derivative of a vector norm w.r.t. a parameter

Question

Let say that we have the vector $V(\theta)=[-2\theta \hspace{0.2cm} \theta^2 \hspace{0.2cm} \theta^3]$, and the elements of $V$ are differentiable functions of $\theta$.

The norm of the vector $V$ equal to $\|V\| = \sqrt{(-2\theta)^2 + (\theta^2)^2 + (\theta^3)^2}$

Is the derivative of the vector norm given by this?:

$$ \frac{\partial \|V\|}{\partial \theta} = \frac{\frac{\partial f}{\partial \theta}}{2 \sqrt{f}}$$ Where $f= (-2\theta)^2 + (\theta^2)^2 + (\theta^3)^2$

Answer 1

Let $g(\theta)=||V(\theta)||=\sqrt{f(\theta)}$. Then, by the chain rule:

$$g'(\theta)=\frac{1}{2 \sqrt{f(\theta)}}f'(\theta).$$