- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.4 Exercises
- Compute the derivatives $F'(t)$ and $F''(t)$ for: $F(t) = (\arcsin t, \arccos t)$
- Recall from section 6.20 that: $$D(\text{arcsin}(t)) = \frac{1}{\sqrt{1 - t^2}}$$if $-1 < t < 1,$ and $$D(\text{arccos}(t)) = \frac{-1}{\sqrt{1 - t^2}}$$if $-1 < t < 1$
- $$F'(t) = \left(\frac{1}{\sqrt{1 - t^2}}, \frac{-1}{\sqrt{1 - t^2}}\right)$$ $$F''(t) = \left(\frac{t}{(1 - t^2)^{3/2}}, \frac{-t}{(1 - t^2)^{3/2}}\right)$$Note: the back-of-book solution has a typo which says that $$F''(t) = \left(\frac{t}{(1 + t^2)^{3/2}}, \frac{-t}{(1 + t^2)^{3/2}}\right)$$