2.2 微分法

1.

(a) $${\frac{-1}{\sqrt{1-x^2}}}$$(b) $${\frac{1}{1 + x^2}}$$

2.

(a) $${x^{2}\sqrt{\frac{x^{3} + 2x + 1}{x^{2} - 3x + 1}}\left(2 + \frac{1}{2}(\frac{3x^2 +2}{x^3 + 2x + 1} - \frac{2x-3}{x^2 - 3x+1}) \right )}$$(b) $${x^{x}(\log{x} + 1)}$$

(c) $${\sin{x^{x}}\left(\log(\sin{x}) + \frac{x\cos{x}}{\sin{x}}\right)}$$(d) $${x^{1/x}\left(\frac{1}{x^2} - \frac{\log{x}}{x^2}\right)}$$

3.

(a) $${-\frac{\cos{t}}{\sin{t}}}$$(b) $${\frac{2t^2 - t^{1/2}}{t^{3/2} + 2}}$$

4.

(a) $${2 x + \frac{5}{2} x^{3/2}}$$(b) $${2 {x^3} {{\sec^{2} (2 x)}} + 3 {x^2} \tan (2 x)}$$(c) $${\frac{x}{\sqrt{1 - {x^2}}} + \sin^{-1} (x)}$$

(d) $${\frac{1 - {x^2}}{\left( 1 + {x^2} \right)^2}}$$(e) $${x \cos (x) + \sin (x)}$$(f) $${\sin^{-1} (x)}$$

(g) $${\frac{2 x}{2 + 2 {x^2} + {x^4}}}$$(h) $${-\frac{\sin ({\sqrt{1 + 2 x}})}{\sqrt{1 + 2 x}}}$$

(i) $${\frac{x^2}{\left( \cos (x) + x \sin (x) \right)^2}}$$(j) $${{e^{2 x}} \left( 2 \cos (x) - \sin (x) \right)}$$(k) $${\frac{1}{\sqrt{A + {x^2}}}}$$

(l) $${2x\cos(x^{2} + 1)}$$(m) $${-\frac{1}{2\sqrt{x+1}}\sin{\sqrt{x+1}}}$$(n) $${e^{\sin{x}}\cos{x}}$$