2.1 Derivatives

1.

(a)

$\displaystyle \frac{\cos(3(x+h)) - \cos(3x)}{h} = \frac{\cos{3x}(\cos{3h} - 1)}{h} - \frac{\sin{3x}\sin{3h}}{h} \rightarrow -3\sin{3x} $

(b)

$\displaystyle \frac{(x+2+h)^{n} - (x+2)^n}{h} = \frac{n(x+2)^{n-1}h}{h} + h(\cdots) \rightarrow n(x+2)^{n-1} $

2.

(a) $df = 4x^3 dx$(b) $df = e^{x} dx$

3.

(a) $\displaystyle{f_{+}^{'}(0) = 1,  f_{-}^{'}(0) = -1}$(b) $\displaystyle{f_{+}^{'}(0) = 0,  f_{-}^{'}(0) = 0}$

(c) $f_{+}^{'}(0) = 1,  f_{-}^{'}(0) = -1$

4.

(a) $\displaystyle{\frac{-3x^2 + 2x + 3}{(x^2 + 1)^2}}$(b) $\displaystyle{\sec{x}\tan{x}}$(c) $\displaystyle{-{\rm cosec}{x}\cot{x}}$ (d) $\displaystyle{-{\rm cosec}^{2}{x}}$

(e) $\displaystyle{x(x+2)e^{x}}$ (f) $\displaystyle{e^{x}(\sin{x} + \cos{x})}$ (g) $\displaystyle{\frac{e^{x}(\sin{x} - \cos{x})}{\sin^{2}{x}}}$