Question

Find \(\mathbf{r}^{\prime}(t)\). $$ \mathbf{r}(t)=\langle\arcsin t, \arccos t, 0\rangle $$

Short answer

The derivative vector \(\mathbf{r}^{\prime}(t)\) is \(\langle\frac{1}{\sqrt{1-t^2}}, -\frac{1}{\sqrt{1-t^2}}, 0\rangle\).

Step-by-step solution

Differentiate the first component

The derivative of the first component of the vector (which is \(\arcsin t\)) is obtained by the derivative of \(\arcsin t\) with respect to \(t\), which is \( \frac{1}{\sqrt{1-t^2}}\).

Differentiate the second component

Next, differentiate the second component of the vector (which is \(\arccos t\)) with respect to \(t\). The derivative of \(\arccos t\) with respect to \(t\) is \(-\frac{1}{\sqrt{1-t^2}}\).

Differentiate the third component

Finally, differentiate the third component, which is a constant (0). The derivative of a constant is always 0.