Advanced Engineering Mathematics

,PRECALCULUS
SIXTH EDITION

,
, Differentiation Rules
d d
1. Constant: c⫽0 2. Constant Multiple: cf (x) ⫽ c f 9(x)
dx dx
d d
3. Sum: ff(x) ⫾ g(x)g ⫽ f 9(x) ⫾ g9(x) 4. Product: f (x)g(x) ⫽ f (x)g9(x) ⫹ g(x) f 9(x)
dx dx
d f(x) g(x)f 9(x) 2 f(x)g9(x) d
5. Quotient: ⫽ 6. Chain: f (g(x)) ⫽ f 9(g(x))g9(x)
dx g(x) fg(x)g 2 dx
d n d
7. Power: x ⫽ nx n 2 1 8. Power: fg(x)g n ⫽ nfg(x)g n 2 1g9(x)
dx dx

Derivatives of Functions
Trigonometric:
d d d
9. sin x ⫽ cos x 10. cos x ⫽ ⫺sin x 11. tan x ⫽ sec 2 x
dx dx dx
d d d
12. cot x ⫽ ⫺csc 2 x 13. sec x ⫽ sec x tan x 14. csc x ⫽ ⫺csc x cot x
dx dx dx
Inverse trigonometric:
d 1 d 1 d 1
15. sin⫺1 x ⫽ 16. cos ⫺1 x ⫽ ⫺ 17. tan⫺1 x ⫽
dx "1 2 x 2 dx "1 2 x 2 dx 1 ⫹ x2
d 1 d 1 d 1
18. cot ⫺1 x ⫽ ⫺ 19. sec ⫺1 x ⫽ 20. csc ⫺1 x ⫽ ⫺
dx 1 ⫹ x2 dx ZxZ"x 2 2 1 dx ZxZ"x 2 2 1
Hyperbolic:
d d d
21. sinh x ⫽ cosh x 22. cosh x ⫽ sinh x 23. tanh x ⫽ sech2 x
dx dx dx
d d d
24. coth x ⫽ ⫺csch2 x 25. sech x ⫽ ⫺sech x tanh x 26. csch x ⫽ ⫺csch x coth x
dx dx dx
Inverse hyperbolic:
d 1 d 1 d 1
27. sinh⫺1 x ⫽ 28. cosh⫺1 x ⫽ 29. tanh⫺1 x ⫽ , ZxZ , 1
dx 2
"x ⫹ 1 dx 2
"x 2 1 dx 1 2 x2
d 1 d 1 d 1
30. coth⫺1 x ⫽ , ZxZ . 1 31. sech⫺1 x ⫽ ⫺ 32. csch⫺1 x ⫽ ⫺
dx 1 2 x2 dx x"1 2 x 2 dx ZxZ"x 2 ⫹ 1

Exponential:
d x d x
33. e ⫽ ex 34. b ⫽ b x(ln b)
dx dx
Logarithmic:
d 1 d 1
35. lnZxZ ⫽ 36. log b x ⫽
dx x dx x(ln b)

Of an integral:
x b b

# # # 0x g(x, t) dt
d d 0
37. g(t) dt ⫽ g(x) 38. g(x, t) dt ⫽
dx a dx a a