.

2.2 Differential Formulas

1.

(a) $${\frac{dy}{dx} = \frac{1}{n}x^{\frac{1}{n} - 1}}$$ (b) $${\frac{dy}{dx} = \frac{1}{2\sqrt{x}}}$$

2.

(a) $${y' = 4008x(x^{2} + 1)^{2003}}$$ (b) $${y' = 6(x^{2} + \frac{1}{x^{2}})^{2}(x - \frac{1}{x^{3}})}$$

(c) $${y' = 6(5x+3)(5x^{2} + 6x + 2)^{2}}$$

3.

(a) $${\frac{dy}{dx} = 2(x-1)}$$ (b) $${\frac{dy}{dx} = - \frac{x}{y}}$$

4.

(a) $${2x\log{x} + x}$$ (b) $${3x^{2}\sin{2x} + 2x^{3}\cos{2x}}$$ (c) $${\frac{2}{\sqrt{1 - 4x^{2}}}}$$ (d) $${\frac{e^{x}}{2\sqrt{e^{x} + 1}}}$$

(e) $${3(\sin{(x+1)})^{2}\cos{(x+1)}}$$ (f) $${\sin^{-1}(2x) + \frac{2x}{\sqrt{1 - 4x^{2}}}}$$