2.1 Derivatives

1.

(a)

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

(b)

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

2.

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

3.

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

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

4.

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

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