1. (c)^′ =  0

2. (x)^′ =  1            (cx)^′ =  c

3. (x²)′= 2x

4. (x³)^′ =  3x²

5. (xⁿ)^′= nx^{n − 1}

6. $\left( \text{lnx} \right)^{'} = \ \frac{1}{x}$

7. (e^x)′ =  e^x

8. (sinx)^′ =  cosx

9. (cosx)^′ =   − sinx

10. $\left( \text{tgx} \right)^{'} = \ \frac{1}{\cos^{2}x}$

11. $\left( \text{ctgx} \right)^{'} = \ \frac{- 1}{\sin^{2}x}$

12. $\left( \sqrt{x} \right)^{'} = \ \frac{1}{2\sqrt{x}}$

13. (a^x)^′ =  a^xlna

14. ${(log}_{a}x)' = \ \frac{1}{\text{xlna}}$

15. $\left( \frac{1}{x} \right)^{'} = \ - \frac{1}{x^{2}}$

Własności pochodnych

W1: [cf(x)]^′ =  cf^′(x)

W2: [f(x)+ −g(x)]^′ =  f^′(x) + −g^′(x)

W3: [f(x)g(x)]^′ =  f^′(x)g(x) +  f(x)g^′(x)

W4: $\left\lbrack \frac{f\left( x \right)}{g\left( x \right)} \right\rbrack^{'} = \ \frac{f^{'}\left( x \right)g\left( x \right) - \ f\left( x \right)g^{'}(x)}{g^{2}(x)}$

W5 (Pochodna funkcji złożonej): [f(g(x))]^′ =  f^′(g(x)) g^′(x)

Własności całki nieoznaczonej:

W1: ∫cf(x)dx = c∫f(x)dx

W2: ∫[f(x)+g(x)]dx = ∫f(x)dx + / − ∫g(x)dx

W3: Całkowanie przez części

∫f^′(x)g(x)dx = f(x)g(x) − ∫f(x)g^′(x)dx

W4: Całkowanie przez podstawianie

∫f(x)dx = ∫f(φ(t))(φ′(t)dt)

Gdzie x=φ(t)-podstawiamy

1. ∫dx = x + c, bo (x+c)^′ = 1

2. $\int_{}^{}{xdx = \frac{1}{2}x^{2} + C,\ bo\left( \frac{x^{2}}{2} + C \right)^{'} = x}$

3. $\int_{}^{}{x^{2}dx = \frac{1}{3}x^{3} + C}$

4. $\int_{}^{}{x^{3}dx = \frac{1}{4}x^{4} + C}$

5. ${\int_{}^{}\mathbf{x}}^{\mathbf{4}}\mathbf{dx =}\frac{\mathbf{x}^{\mathbf{n + 1}}}{\mathbf{n + 1}}\mathbf{+ C,\ n \neq 1}$

6. $\int_{}^{}{\frac{\text{dx}}{x} = lnx + C}$

7. ∫sinxdx = −cosx + C

8. ∫cosxdx = sinx + C

9. ∫e^xdx = e^x + C

10. $\int_{}^{}{a^{x}dx = \frac{a^{x}}{\text{lna}}}$

11. $\int_{}^{}{\frac{\text{dx}}{x^{2}} = - \frac{1}{x} + C}$

12. $\int_{}^{}{\frac{\text{dx}}{2\sqrt{x}} = \sqrt{x} + C}$

13. $\int_{}^{}{\frac{\text{dx}}{\sqrt{1 - x^{2}}} = arcsinx + C}$

14. $\int_{}^{}{\frac{- dx}{\sqrt{1 - x^{2}}} = arccosx + C}$

15. $\int_{}^{}{\frac{\text{dx}}{1 + x^{2}} = arctgx + C}$

16. $\int_{}^{}{\frac{f^{'}(x)}{f(x)}\text{dx}} = \ln\left( f\left( x \right) \right) + C$

f(x)=t, f’(x)dx = dt $\int_{}^{}{\frac{\text{dt}}{t} = \ln\left( t \right) + C}$

17. $\int_{}^{}{\frac{\text{dx}}{x^{2}a^{2}} = \frac{1}{a}arctg(\frac{x}{a})} + C$

18.$\int_{}^{}{e^{\text{dx}}\text{dx}} = \frac{1}{\propto}e^{\propto x} + C$

19.$\int_{}^{}{\frac{f^{'}(x)}{\sqrt{f(x)}}\text{dx}} = 2\sqrt{f(x)} + C$

6’.$\int_{}^{}\frac{\text{dx}}{x - a}\ \left\| x - a = t,\ dx = dt \right\|$

$$\int_{}^{}{\frac{\text{dt}}{t} = \ln\left( t \right) + C = \ln\left( x - a \right) + C}$$

20.∫tgxdx = −ln(cosx) + C

21.∫ctgxdx = ln(sinx) +  C

22.$\int_{}^{}\frac{\text{dx}}{\cos^{2}x} = tgx + C$

23.$\int_{}^{}\frac{\text{dx}}{\sin^{2}x} = - ctgx + C$