Differentiating $\vec{r}(t)=(3\sin\frac{t}{t_0},4\frac{t}{t_0},3 \cos \frac{t}{t_0}); \space \space t\in \Bbb R$

Question

I want to differentiate the following vector with respect to $t$

$$\vec{r}(t)=(3\sin\frac{t}{t_0},4\frac{t}{t_0},3 \cos \frac{t}{t_0}); \space \space t\in \Bbb R$$

Can I treat $t_0$ like a constant so that

$$\dot{\vec{r}}(t)=(\frac{3}{t_0} \cos (\frac{t}{t_0}), \frac{4}{t_0},-\frac{3}{t_0}\sin(\frac{t}{t_0}))$$

or am I differentiating wrong here?

Answer 1

You are exactly correct. $t_0$ is held constant, and you just take derivative in each direction separately.

Answer 2

You can treat it as a constant, yes. It is independent of $t$ (probably, it usually is the case with definitions like this...)