Spring 2019 Math 152
Formulas from Calculus I
courtesy: Amy Austin
Derivatives
1. d
dxxn=nxn−1
2. d
dx ln x=1
x
3. d
dx ln(g(x)) = g′(x)
g(x)
4. d
dxex=ex
5. d
dxax=axln a
6. d
dxeg(x)=g′(x)eg(x)
7. d
dxag(x)=g′(x)ag(x)ln a
8. d
dx cos−1x=−1
√1−x2
9. d
dx sin−1x=1
√1−x2
10. d
dx tan−1x=1
1 + x2
11. d
dx sin x= cos x
12. d
dx cos x=−sin x
13. d
dx tan x= sec2x
14. d
dx sec x= sec xtan x
15. d
dx csc x=−csc xcot x
16. d
dx cot x=−csc2x
17. Product Rule: d
dxgh =g′h+gh′
18. Quotient Rule: d
dx
g
h=g′h−gh′
h2
19. Chain Rule: d
dxf(g(x)) = f′(g(x))g′(x)
Integrals
20. Zxndx =xn+1
n+ 1 +C, if n6=−1
21. Zexdx =ex+C
22. Zaxdx =ax
ln a+C
23. Z1
xdx = ln |x|+C
24. Z1
1 + x2dx = arctan x+C
25. Z1
a2+x2dx =1
aarctan x
a+C
26. Z1
√1−x2dx = arcsin x+C
27. Zcos x dx = sin x+C
28. Zsin x dx =−cos x+C
29. Zsec xtan x dx = sec x+C
30. Zsec2x dx = tan x+C
31. Zcsc xcot x dx =−csc x+C
32. Zcsc2x dx =−cot x+C
Logarithm Rules
33. ln P Q = ln P+ ln Q
34. ln P
Q= ln P−ln Q
35. ln Pr=rln P
Useful Trig Identities
36. cos2x+ sin2x= 1
37. tan2x+ 1 = sec2x
38. cos2x=1
2[1 + cos 2x]
39. sin2x=1
2[1 −cos 2x]
40. sin 2x= 2 sin xcos x
41. cos 2x= 1 −2 sin2x= 2 cos2x−1
