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preface to fifteenth edition

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2.104 SECTION 2<br />

TABLE 2.23 Derivatives and Differentiation<br />

Rules for differentiation.<br />

From Baumeister and Marks, Standard Handbook for Mechanical Engineers, 7thed., McGraw-Hill Book<br />

Company, New York (1967); by permission.<br />

To find the derivative of a given function at a given point: (1) If the function is given only by a curve,<br />

measure graphically the slope of the tangent at the point in question; (2) if the function is given by a mathematical<br />

expression, use the following rules for differentiation. These rules give, directly, the differential, dy, in terms of<br />

dx; <strong>to</strong> find the derivative, dy/dx, divide through by dx.<br />

Here u, v, w, ...represents any functions of a variable x, or may themselves be independent variables. a<br />

is a constant which does not change in values in the same discussion; e 2.71828.<br />

1. d(a u) du 2. d(au) adu<br />

3. d(u v w ···) du dv dw ···<br />

4. d(uv) udv vdu<br />

du dv dw<br />

5. d(uvw ...) (uvw ...) ··· <br />

u v w<br />

u v du udv<br />

m m1<br />

6. d <br />

7. d(u ) mu du<br />

v v 2<br />

2 3 2<br />

Thus, d(u ) 2u du; d(u ) 3u du; etc.<br />

du<br />

1 du<br />

8. d pu <br />

9. d <br />

2<br />

2pu<br />

u u<br />

u u<br />

u u<br />

10. d(e ) e du 11. d(a ) (ln a)a du<br />

du<br />

du du<br />

12. d ln u 13. d log10 u (log10<br />

e) (0.4343 ...)<br />

u u<br />

u<br />

14. d sin u cos udu 15. d csc u cot u csc udu<br />

16. d cos u sin udu 17. d sec u tan u sec udu<br />

2<br />

2<br />

18. d tan u sec udu 19. d cot u csc udu<br />

du<br />

du<br />

20. d sin1<br />

u <br />

21. d csc1<br />

u <br />

p1 u 2<br />

u u2<br />

p 1<br />

du<br />

du<br />

22. d cos1<br />

u <br />

23. d sec1<br />

u <br />

p1 u 2<br />

u u2<br />

p 1<br />

du<br />

du<br />

1<br />

1<br />

24. d tan u <br />

25. d cot u <br />

1 u 2<br />

1 u 2<br />

2 du<br />

26. d ln sin u cot udu<br />

27. d ln tan u sin 2u<br />

2 du<br />

28. d ln cos u tan udu<br />

29. d ln cot u sin 2u<br />

30. d sinh u cosh udu 31. d csch u csch u coth udu<br />

32. d cosh u sinh udu 33. d sech u sech u tanh udu<br />

34. d tanh u sech2<br />

udu 35. d coth u csch2<br />

udu<br />

du<br />

du<br />

1<br />

1<br />

36. d sinh u 37. d csch u <br />

2<br />

pu 1<br />

2<br />

upu 1<br />

du<br />

du<br />

1<br />

1<br />

38. d cosh u 39. d sech u <br />

2<br />

pu 1<br />

up1 u 2<br />

du<br />

du<br />

1<br />

1<br />

40. d tanh u <br />

41. d coth u <br />

1 u 2<br />

1 u 2<br />

v v1<br />

42. d(u ) (u )(u ln udv vdu)

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