Exercises

1. For each of the following functions, calculate both first and second derivatives.

   \(y=e^{-t}\sin 5t\)	\(z=x^2\, 3^x\)
   \(w=\cos^3\, 2t\)	\(w=\cos (2t^3)\)
   \(z=\sin^{-1}x\,\tan 2x\)	\(w=\sin^{-1}x\,\,\,\,\tan^{-1}\,2x\)
   \(y=\frac{e^{2x}}{\sin 5x}\)	\(y=\frac{e^{2x}}{\cos 5x}\)

2. Calculate each of the following derivatives.

   \(\frac{d}{dx}\,e^{-2x}\,\cos 5x\)	\(\frac{d^2}{d \theta^2}\,e^{-2\theta}\,\cos 5\theta\)
   \(\frac{d}{dx}\,\left(\ln x^2\right)\,\cos 2x\)	\(\frac{d^2}{d \theta^2}\,\left(\ln\theta^2\right)\,\cos 2\theta\)
   \(\frac{d}{dt}\,3^{2t}\)	\(\frac{d^2}{d \theta^2}\,3^{2\theta}\)

3. Find the derivative of each of the following functions.

   \(z=7\ln(3+2t)\)	\(z=\frac{1}{2t+1}\)
   \(y=x^6+4x^2+6x^{-2}\)	\(z=\frac{7}{3+2t}\)
   \(y=2e^{5x}\)	\(y=\frac{\sin 2u}{\cos 5u}\)
   \(y=(\sin 2t) (\cos 5t)\)	\(z=\left(2t+1\right)^{12}\)
   \(y=(\sin^{-1} 2t) (\cos 5t)\)	\(u=\tan^3\, 2x\)

4. Find the derivative of each of the following functions.

   \(y=3\sin t \cos 3t\)	\(z=\frac{1}{(2t+1)^2}\)
   \(u=\cos^2\, 3t\)	\(z=\frac{7}{(3+2t)^3}\)
   \(v=\cos(3t^2)\)	\(y=x^7+3x^2+6\)
   \(y=\sin 2t - \cos 5t\)	\(z=2^{5x}\)
   \(y=\tan 2t - \cos 5t\)	\(v=\sin^{-1}\,5y\)