More review for quiz 2

Here are a few more problems like our quiz this Friday.

Problems

1. Use the differentiation rules to compute the derivatives of the following functions.
   1. \(\displaystyle f(x) = x\tan(x)\)
   2. \(\displaystyle f(x) = \frac{\sqrt{x}}{x^2 + 1}\)

2. Use the fact that \(\displaystyle \lim_{\theta\to0} \sin(\theta)/\theta = 1\) to compute \[ \lim_{x\to0} \frac{\tan(4x)}{3x}. \]

3. Sketch the graph of \[f(x) = -\sin\left(\pi x\right).\]
   Be sure to clearly indicate the \(x\) and \(y\) intercepts, as well as the maximum and minimum values.

4. Supposing that \(f\) and \(g\) are differentiable functions, use the definition of the derivative to show that \[\frac{d}{dx} f(x)+3g(x) = f'(x)+3g'(x).\]