- 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: $F(t) = \log(1 + t^2)\mathbf{i} + \arctan t\mathbf{j} + \frac{1}{1+t^2}\,\mathbf{k}$
- Recall from section 6.20 that $$D(\text{arctan}(t)) = \frac{1}{1 + t^2}$$for all real $t$
- $$F'(t) = \left(\frac{2t}{1 + t^2}, \frac{1}{1 + t^2}, \frac{-2t}{(1 + t^2)^2}\right)$$$$F''(t) = \left(\frac{(2 - 2t^2)}{(1 + t^2)^2}, \frac{-2t}{(1 + t^2)^2}, \frac{6t^2 - 2}{(1 + t^2)^3}\right)$$