Calculus: 50 derivatives, integrals and limits

All 50 cards

1. 1
   $\dfrac{d}{dx}\,c$ (a constant)
   $0$
2. 2
   $\dfrac{d}{dx}\,x^n$
   $n\,x^{n-1}$

   The power rule.
3. 3
   $\dfrac{d}{dx}\,e^x$
   $e^x$
4. 4
   $\dfrac{d}{dx}\,a^x$
   $a^x \ln a$
5. 5
   $\dfrac{d}{dx}\,\ln x$
   $\dfrac{1}{x}$
6. 6
   $\dfrac{d}{dx}\,\log_a x$
   $\dfrac{1}{x \ln a}$
7. 7
   $\dfrac{d}{dx}\,\sin x$
   $\cos x$
8. 8
   $\dfrac{d}{dx}\,\cos x$
   $-\sin x$
9. 9
   $\dfrac{d}{dx}\,\tan x$
   $\sec^2 x$
10. 10
    $\dfrac{d}{dx}\,\cot x$
    $-\csc^2 x$
11. 11
    $\dfrac{d}{dx}\,\sec x$
    $\sec x \tan x$
12. 12
    $\dfrac{d}{dx}\,\csc x$
    $-\csc x \cot x$
13. 13
    $\dfrac{d}{dx}\,\arcsin x$
    $\dfrac{1}{\sqrt{1 - x^2}}$
14. 14
    $\dfrac{d}{dx}\,\arccos x$
    $-\dfrac{1}{\sqrt{1 - x^2}}$
15. 15
    $\dfrac{d}{dx}\,\arctan x$
    $\dfrac{1}{1 + x^2}$
16. 16
    $\dfrac{d}{dx}\,\sinh x$
    $\cosh x$
17. 17
    $\dfrac{d}{dx}\,\cosh x$
    $\sinh x$
18. 18
    $\dfrac{d}{dx}\,\sqrt{x}$
    $\dfrac{1}{2\sqrt{x}}$
19. 19
    $\dfrac{d}{dx}\,\dfrac{1}{x}$
    $-\dfrac{1}{x^2}$
20. 20
    Product rule
    $(fg)' = f'g + fg'$
21. 21
    Quotient rule
    $\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$
22. 22
    Chain rule
    $\dfrac{d}{dx}\,f(g(x)) = f'(g(x))\,g'(x)$
23. 23
    Definition of the derivative
    $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
24. 24
    $\displaystyle\lim_{x \to 0} \frac{\sin x}{x}$
    $1$
25. 25
    $\displaystyle\lim_{x \to 0} \frac{1 - \cos x}{x}$
    $0$
26. 26
    $\displaystyle\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$
    $e$
27. 27
    $\displaystyle\lim_{x \to 0} \frac{e^x - 1}{x}$
    $1$
28. 28
    L’Hôpital’s rule
    For $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$ forms: $\displaystyle\lim \frac{f}{g} = \lim \frac{f'}{g'}$, if the right side exists
29. 29
    $\displaystyle\int x^n\,dx$ for $n \neq -1$
    $\dfrac{x^{n+1}}{n + 1} + C$
30. 30
    $\displaystyle\int \frac{1}{x}\,dx$
    $\ln\lvert x\rvert + C$
31. 31
    $\displaystyle\int e^x\,dx$
    $e^x + C$
32. 32
    $\displaystyle\int a^x\,dx$
    $\dfrac{a^x}{\ln a} + C$
33. 33
    $\displaystyle\int \sin x\,dx$
    $-\cos x + C$
34. 34
    $\displaystyle\int \cos x\,dx$
    $\sin x + C$
35. 35
    $\displaystyle\int \sec^2 x\,dx$
    $\tan x + C$
36. 36
    $\displaystyle\int \frac{1}{1 + x^2}\,dx$
    $\arctan x + C$
37. 37
    $\displaystyle\int \frac{1}{\sqrt{1 - x^2}}\,dx$
    $\arcsin x + C$
38. 38
    $\displaystyle\int \tan x\,dx$
    $-\ln\lvert\cos x\rvert + C$

    Equivalently $\ln\lvert\sec x\rvert + C$.
39. 39
    $\displaystyle\int \sec x\,dx$
    $\ln\lvert\sec x + \tan x\rvert + C$
40. 40
    $\displaystyle\int \ln x\,dx$
    $x \ln x - x + C$

    Integrate by parts with u = ln x, dv = dx.
41. 41
    Integration by parts
    $\displaystyle\int u\,dv = uv - \int v\,du$
42. 42
    Substitution rule
    $\displaystyle\int f(g(x))\,g'(x)\,dx = \int f(u)\,du$ with $u = g(x)$
43. 43
    Fundamental theorem of calculus, part 1
    $$\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$$

    For f continuous on the interval.
44. 44
    Fundamental theorem of calculus, part 2
    $$\int_a^b f(x)\,dx = F(b) - F(a)$$

    Where $F$ is any antiderivative of $f$.
45. 45
    $\displaystyle\int_0^1 x^2\,dx$
    $\dfrac{1}{3}$
46. 46
    $\displaystyle\int_{-\infty}^{\infty} e^{-x^2}\,dx$
    $\sqrt{\pi}$

    The Gaussian integral.
47. 47
    Mean value theorem
    If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, some $c$ in $(a, b)$ has $f'(c) = \dfrac{f(b) - f(a)}{b - a}$
48. 48
    Second derivative test
    If $f'(c) = 0$: $f''(c) > 0$ means a local minimum, $f''(c) < 0$ a local maximum
49. 49
    Taylor series of $e^x$
    $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$
50. 50
    Taylor series of $\sin x$
    $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$