微分常用公式

1. $$(f(x) + g(x))' = f'(x) + g'(x)$$

2. $$(f(x) - g(x))' = f'(x) - g'(x)$$

3. $$(cf(x))' = c f'(x) \quad \text{（c為常數）}$$

4. $$(f(x)g(x))' = f'(x)g(x) + f(x)g'(x) \quad \text{（乘法法則）}$$

5. $$\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \quad \text{（除法法則）}$$

6. $$(f(g(x)))' = f'(g(x)) \cdot g'(x) \quad \text{（鏈式法則）}$$

7. $$\frac{d}{dx}(x^n) = nx^{n-1}$$

8. $$\frac{d}{dx}(\sin x) = \cos x$$

9. $$\frac{d}{dx}(\cos x) = -\sin x$$

10. $$\frac{d}{dx}(\tan x) = \sec^2 x$$

11. $$\frac{d}{dx}(e^x) = e^x$$

12. $$\frac{d}{dx}(\ln x) = \frac{1}{x}$$

積分常用公式

1. $$\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$

2. $$\int \frac{1}{x} dx = \ln|x| + C$$

3. $$\int e^x dx = e^x + C$$

4. $$\int \sin x , dx = -\cos x + C$$

5. $$\int \cos x , dx = \sin x + C$$

6. $$\int \sec^2 x , dx = \tan x + C$$