Chapter 3.3, Problem 64ES

(a) Use part (c) of Exercise $30$ in Section $1.7$ and the chain rule to show that

$\frac{d}{dx}\left[{\csc }^{-1}x\right]=-\frac{1}{\left|x\right|\sqrt{x^2-1}}$

for $1<\left|x\right|$ .

(b) Conclude from part (a) that

$\frac{d}{dx}\left[{\csc }^{-1}u\right]=-\frac{1}{\left|u\right|\sqrt{u^2-1}}\frac{du}{dx}$

for $1<\left|u\right|$ .

(c) Use Equation (8) in Section $1.7$ and parts (b) and (c) of Exercise $30$ in that section to show that if $\left|x\right|\ge 1$ , then ${\sec }^{-1}x+{\csc }^{-1}x=\pi /2$ . Conclude from part (a) that

$\frac{d}{dx}\left[{\sec }^{-1}x\right]=\frac{1}{\left|x\right|\sqrt{x^2-1}}$

(d) Conclude from part (c) that

$\frac{d}{dx}\left[{\sec }^{-1}u\right]=\frac{1}{\left|u\right|\sqrt{x^2-1}}\frac{du}{dx}$