3.8: Derivatives of Inverse Functions

1. Find $\arcsin\left(\dfrac{1}{2}\right)$.
    
2. Find $\arctan(-1)$.
    
3. Find $\sec^{-1}(-2)$.
    
4. Find $\cot^{-1}(0)$.
    
5. Find $\cos^{-1}\left(\cos\left(\dfrac{\pi}{4}\right)\right)$.
    
6. Find $\arcsin\left(\sin\left(\dfrac{2\pi}{3}\right)\right)$.
    
7. Find $\sec^{-1}\left(\sec\left(-\dfrac{\pi}{6}\right)\right)$.
    
8. Show that $\dfrac{d}{dx}\arcsin x = \dfrac{1}{\sqrt{1 - x^2}}$.
    
9. Show that $\dfrac{d}{dx}\cos^{-1} x = \dfrac{-1}{\sqrt{1 - x^2}}$.
    
10. Show that $\dfrac{d}{dx}\arctan x = \dfrac{1}{1 + x^2}$.
     
11. Show that $\dfrac{d}{dx}\cot^{-1} x = \dfrac{-1}{1 + x^2}$.
     
12. Show that $\dfrac{d}{dx}\text{arcsec} x = \dfrac{1}{|x|\sqrt{x^2 - 1}}$.
     
13. Show that $\dfrac{d}{dx}\csc^{-1} x = \dfrac{-1}{|x|\sqrt{x^2 - 1}}$.
     
14. Given $y = \arcsin(x^2)$, find $\dfrac{dy}{dx}$.
     
15. Given $y = \cos^{-1}\Bigl(\sin (x^3)\Bigr)$, find $\dfrac{dy}{dx}$.
     
16. Given $y = \sec^{-1}\left(\dfrac{1}{x}\right)$, find $\dfrac{dy}{dx}$.
     
17. Given $y = \arccos(e^x) \cdot \arcsin(e^x)$, find $\dfrac{dy}{dx}$.
     
18. Given $y = \ln \Bigl((\sin^{-1} x)^2\Bigr)$, find $\dfrac{dy}{dx}$.
     
19. Given $y = \tan^{-1} \sqrt{9 - x^2}$, find $\dfrac{dy}{dx}$.