基本求导公式

$$1. \ (x^\alpha)^{\prime} \ = \ ax^{a \ - \ 1}$$

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$$2. \ (a^x)^{\prime} \ = \ (a^x)\ln{a} \ \ \ (a>0, \ a \neq 1)$$

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$$3. \ (e^x)^{\prime} \ = \ e^x$$

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$$4. \ (log_ax)^{\prime} \ = \ \frac{1}{x\ln{a}} \ \ \ (a>0,a \neq 1)$$

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$$5. \ (\ln{x})^{\prime} \ = \ \frac{1}{x}$$

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$$6. \ (sinx)^{\prime} \ = \ cosx$$

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$$7. \ (cosx)^{\prime} \ = \ -sinx$$

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$$8. \ (\arcsin{x})^{\prime} \ = \ \frac{1}{\sqrt{1 \ - \ x^2}}$$

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$$9. \ (\arccos{x})^{\prime} \ = \ -\frac{1}{\sqrt{1 \ - \ x^2}}$$

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$$10. \ (\tan{x})^{\prime} \ = \ \sec{^2x}$$

证明：

$(\tan{x})^{\prime}$

$= \ (\frac{\sin{x}}{\cos{x}})^{\prime}$

$= \ \frac{\cos{x} \cos{x} \ - \ \sin{x}(-\sin{x})}{\cos{^2x}}$

$= \ \frac{1}{\cos{^2x}}$

$= \ \sec{^2x}$

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$$11. \ (\cot{x})^{\prime} \ = \ -\csc{^2x}$$

证明：

$(\cot{x})^{\prime}$

$= \ (\frac{\cos{x}}{\sin{x}})^{\prime}$

$= \ (\frac{-\sin{x} \sin{x} \ - \ \cos{x} \cos{x}}{\sin{^2x}})$

$= \ -\frac{1}{\sin{^2x}}$

$= \ -\csc{^2x}$

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$$12. \ (\arctan{x})^{\prime} \ = \ \frac{1}{1 \ + \ x^2} \ \ \ (暂时不会证明)$$

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$$13. \ (arccot{x})^{\prime} \ = \ -\frac{1}{1 \ + \ x^2}$$

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$$14. \ (\sec{x})^{\prime} \ = \ \sec{x} \tan{x}$$

证明：

$(\sec{x})^{\prime}$

$= \ (\frac{1}{\cos{x}})^{\prime}$

$= \ \frac{0 \ - \ (-\sin{x})}{\cos{^2x}}$

$= \ \frac{sin{x}}{cos{x}} \frac{1}{\cos{x}}$

$= \ \sec{x} \tan{x}$

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$$15. \ (\csc{x})^{\prime} \ = \ -\csc{x} \cot{x}$$

证明：

$(\csc{x})^{\prime}$

$= \ (\frac{1}{\sin{x}})^{\prime}$

$= \ \frac{0 \ - \ \cos{x}}{\sin{^2x}}$

$= \ -\frac{\cos{x}}{\sin{x}} \frac{1}{\sin{x}}$

$= \ -\csc{x} \ \cot{x}$

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$$16. \ [\ln{(x \ + \ \sqrt{x^2 \ + \ 1}})]^{\prime} \ = \ \frac{1}{\sqrt{x^2 \ + \ 1}}$$

证明：

$[\ln{(x \ + \ \sqrt{x^2 \ + \ 1}})]^{\prime}$

$= \ \frac{1 \ + \ \frac{1}{2}\frac{2x}{\sqrt{x^2 \ + \ 1}}}{x \ + \ \sqrt{x^2 \ + \ 1}}$

$= \ \frac{1 \ + \ \frac{x}{\sqrt{x^2 \ + \ 1}}}{x \ + \ \sqrt{x^2 \ + \ 1}}$

$= \ \frac{\frac{x \ + \ \sqrt{x^2 \ + \ 1}}{\sqrt{x^2 \ + \ 1}}}{x \ + \ \sqrt{x^2 \ + \ 1}}$

$= \ \frac{1}{\sqrt{x^2 \ + \ 1}}$

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$$17. \ [\ln{(x \ + \ \sqrt{x^2 \ - \ 1}})]^{\prime} \ = \ \frac{1}{\sqrt{x^2 \ - \ 1}}$$