# Vector Derivative and Tangent – Calculating Derivative and a derivative size of a vector function – Exercise 3820

Exercise

Given the vector function in the parametric presentation

$$\vec{r}(t)=\cos (2t)\vec{i}+\sin (2t)\vec{j}+t^2\vec{k}$$

Calculate

$$\frac{d\vec{r}}{dt},|\frac{d\vec{r}}{dt}|,\frac{d\vec{|r|}}{dt}$$

$$\frac{d\vec{r}}{dt}=-2\sin (2t)\vec{i}+2\cos (2t)\vec{j}+2t\vec{k}$$
$$|\frac{d\vec{r}}{dt}|=2\sqrt{1+t^2}$$
$$\frac{d\vec{|r|}}{dt}=\frac{2t^3}{\sqrt{1+t^4}}$$