Page 80 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 4] DERIVATIVES 71

DERIVATIVES OF ELEMENTARY FUNCTIONS
In the following we assume that u is a differentiable function of x;if u ¼ x, du=dx ¼ 1. The inverse
functions are defined according to the principal values given in Chapter 3.
d d 1 1 du
1. ðCÞ¼ 0 16. cot u ¼
2
dx dx 1 þ u dx

d n n 1 du d 1 1 du þ if u > 1
2. u ¼ nu 17. sec u ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi
2
dx dx dx u u 1 dx if u < 1

d du d 1 1 du if u > 1
3. sin u ¼ cos u 18. csc u ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi
2
dx dx dx u u 1 dx þ if u < 1
d du d du
4. cos u ¼ sin u 19. sinh u ¼ cosh u
dx dx dx dx
d 2 du d du
5. tan u ¼ sec u 20. cosh u ¼ sinh u
dx dx dx dx
d 2 du d 2 du
6. cot u ¼ csc u 21. tanh u ¼ sech u
dx dx dx dx
d du d 2 du
7. sec u ¼ sec u tan u 22. coth u ¼ csch u
dx dx dx dx
d du d du
8. csc u ¼ csc u cot u 23. sech u ¼ sech u tanh u
dx dx dx dx
d log e du d du
a
9. log u ¼ a > 0; a 6¼ 1 24. csch u ¼ csch u coth u
a
dx u dx dx dx
d d 1 du d 1 1 du
10. log u ¼ ln u ¼ 25. sinh u ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi
e
dx dx u dx dx 1 þ u 2 dx
d u u du d 1 1 du
11. a ¼ a ln a 26. cosh u ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi
dx dx dx u 1 dx
2
d u u du d 1 1 du
12. e ¼ e 27. tanh u ¼ ; juj < 1
2
dx dx dx 1 u dx
d 1 du d 1 du
13. sin 1 u ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi 28. coth 1 u ¼ ; juj > 1
2
dx 1 u dx dx 1 u dx
2
d 1 du d 1 du
14. cos 1 u ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi 29. sech 1 u ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi
dx 1 u dx dx u 1 u dx
2
2
d 1 du d 1 du
15. tan 1 u ¼ 30. csch 1 u ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi
2
dx 1 þ u dx dx u u þ 1 dx
2
HIGHER ORDER DERIVATIVES
If f ðxÞ is differentiable in an interval, its derivative is given by f ðxÞ, y or dy=dx, where y ¼ f ðxÞ.If
0
0
2
d dy d y
f ðxÞ is also differentiable in the interval, its derivative is denoted by f ðxÞ, y 00 or dx dx ¼ dx 2 .
00
0
n
d y
Similarly, the nth derivative of f ðxÞ,ifit exists, is denoted by f ðnÞ ðxÞ, y ðnÞ or , where n is called the
dx n
order of the derivative. Thus derivatives of the first, second, third, . . . orders are given by f ðxÞ, f ðxÞ,
00
0
f ðxÞ; ... .
000
Computation of higher order derivatives follows by repeated application of the differentiation rules
given above.