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Calculus I Rules for Derivatives

In Calculus I, the primary focus is the derivative of a function. The derivative is used to determine rates of change. There are many rules for derivatives that are based on concepts from previous math. These rules include taking derivatives of products, exponents, trigonometry functions as well as many others.

Some of the rules used in calculus are derived from other rules. This is especially the case for certain rules for derivatives. Rules that contain a function are based on the chain rule even though they might fit a different rule better. The derivative rule for the tangent function can be derived from the power rule using sines and cosines.

1. Constant Rule       d
dx
[c] = 0

2. Constant Multiple Rule       d
dx
[cu] = cu'

3. Sum/Difference Rule       d
dx
[u ± v] = u' ± v'

4. Product Rule       d
dx
[uv] = u'v + uv'

5. Quotient Rule       d
dx
[ u
v
]  =  u'v − uv'
v2

6. Very Simple Power Rule       d
dx
[x] = 1

7. Simple Power Rule       d
dx
[xn] = nxn−1

8. General Power Rule       d
dx
[un] = nun−1u'

9. Chain Rule       d
dx
[ƒ(u)] = ƒ'(u)u'

10. Simple Absolute Value Derivative Rule       d
dx
[ |x| ] =  x
|x|

11. General Absolute Value Derivative Rule       d
dx
[ |u| ] =  u
|u|
u'

12. Simple Exponential Derivative Rule       d
dx
[ex] = ex

13. General Exponential Derivative Rule       d
dx
[eu] = euu'

14. Simple Different Exponential Base Derivative Rule       d
dx
[ax] = (ln a)ax

15. General Different Exponential Base Derivative Rule       d
dx
[au] = (ln a)auu'

16. Simple Logarithm Derivative Rule       d
dx
[ln x] =  1
x

17. General Logarithm Derivative Rule       d
dx
[ln u] =  u'
u

18. Simple Different Base Logarithm Derivative Rule       d
dx
[lnax] =  1
(ln a)x

19. General Different Base Logarithm Derivative Rule       d
dx
[lnau] =  u'
(ln a)u

20. Simple Derivative of Sine Rule       d
dx
[sin x] = cos x

21. Simple Derivative of Cosine Rule       d
dx
[cos x] = −sin x

22. Simple Derivative of Tangent Rule       d
dx
[tan x] = sec2 x

23. Simple Derivative of Cotangent Rule       d
dx
[cot x] = −csc2 x

24. Simple Derivative of Secant Rule       d
dx
[sec x] = sec x tan x

25. Simple Derivative of Cosecant Rule       d
dx
[csc x] = −csc x cot x

26. General Derivative of Sine Rule       d
dx
[sin u] = u'(cos u)

27. General Derivative of Cosine Rule       d
dx
[cos u] = −u'(sin u)

28. General Derivative of Tangent Rule       d
dx
[tan u] = u'(sec2 u)

29. General Derivative of Cotangent Rule       d
dx
[cot u] = −u'(csc2 x)

30. General Derivative of Secant Rule       d
dx
[sec u] = u'(sec u tan u)

31. General Derivative of Cosecant Rule       d
dx
[csc u] = −u'(csc u cot u)

32. Simple Arcsine Derivative Rule       d
dx
[arcsine x] =  1
√(1−x2)

33. Simple Arccosine Derivative Rule       d
dx
[arccosine x] =  −1
√(1−x2)

34. Simple Arctangent Derivative Rule       d
dx
[arctangent x] =  1
1+x2

35. Simple Arccotangent Derivative Rule       d
dx
[arccotangent x] =  −1
1+x2

36. Simple Arcsecant Derivative Rule       d
dx
[arcsecant x] =  1
|x|√(x2−1)

37. Simple Arccosecant Derivative Rule       d
dx
[arccosecant x] =  −1
|x|√(x2−1)

38. General Arcsine Derivative Rule       d
dx
[arcsine u] =  u'
√(1−u2)

39. General Arccosine Derivative Rule       d
dx
[arccosine u] =  −u'
√(1−u2)

40. General Arctangent Derivative Rule       d
dx
[arctangent u] =  u'
1+u2

41. General Arccotangent Derivative Rule       d
dx
[arccotangent u] =  −u'
1+u2

42. General Arcsecant Derivative Rule       d
dx
[arcsecant u] =  u'
|u|√(u2−1)

43. General Arccosecant Derivative Rule       d
dx
[arccosecant u] =  −u'
|u|√(u2−1)

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