Exercises

1. Calculate each of the following derivatives.
   

   \(\frac{d}{dt}\,\tan\,5t\)	\(\frac{d}{dx}\,\sin\,5x\,\tan\,5x\)
   \(\frac{d}{dx}\,\tan^{-1}\,5x\)	\(\frac{d}{dt}\,\cos\,5t\,\tan\,5t\)
   \(\frac{d}{dt}\,e^{-t}\,\tan\,5t\)	\(\frac{d}{dx}\,\sin^{-1}\,5x\,\tan\,5x\)
   \(\frac{d}{dx}\,e^{-2x}\,\tan^{-1}\,5x\)	\(\frac{d}{dt}\,\sin\,5t\,\tan^{-1}\,5t\)

2. Calculate each of the following second derivatives. [Recall that \(\frac{d}{dt}\sec\,t= \sec\,t\,\tan\,t\).].

   \(\frac{d^2}{dt^2}\tan\,5t\)	\(\frac{d^2}{d t^2}\sin^{-1}5t\)
   \(\frac{d^2}{d t^2}\tan^{-1}5t\)	\(\frac{d^2}{d \theta^2}\cos\,5\theta\,\tan\,5\theta\)

3. 1. What is the domain of the inverse tangent function? What is its range?
   2. Graph the arctangent function. (The Graph tool will accept either atan or arctan as the name.)
   
   
   3. Graph the derivative of the arctangent function.
   4. What is the steepest slope on the graph of the arctangent function, and where does it occur?