2.2 Differentiation formulas

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

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

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

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

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

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

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

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

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