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  1. and it turns out that it’s actually fairly simple to differentiate a function composition using the Chain Rule.[1]
  2. In general, we don’t really do all the composition stuff in using the Chain Rule.[1]
  3. In general, this is how we think of the chain rule.[1]
  4. First, there are two terms and each will require a different application of the chain rule.[1]
  5. For the chain rule in probability theory, see Chain rule (probability) .[2]
  6. In calculus, the chain rule is a formula to compute the derivative of a composite function.[2]
  7. The chain rule may also be rewritten in Leibniz's notation in the following way.[2]
  8. The chain rule seems to have first been used by Gottfried Wilhelm Leibniz.[2]
  9. Chain rule, in calculus, basic method for differentiating a composite function.[3]
  10. The chain rule has been known since Isaac Newton and Leibniz first discovered the calculus at the end of the 17th century.[3]
  11. Recall that we used the ordinary chain rule to do implicit differentiation.[4]
  12. The chain rule has a particularly simple expression if we use the Leibniz notation for the derivative.[5]
  13. This is a quotient with a constant numerator, so we could use the quotient rule, but it is simpler to use the chain rule.[5]
  14. And it's called the chain rule.[6]
  15. And to do this, I'm going to use the chain rule.[6]
  16. I'm going to use the chain rule, and the chain rule comes into play every time, any time your function can be used as a composition of more than one function.[6]
  17. And so there we've applied the chain rule.[6]
  18. Now that we have derived a special case of the chain rule, we state the general case and then apply it in a general form to other composite functions.[7]
  19. The Chain Rule Let \(f\) and \(g\) be functions.[7]
  20. This rule is called the chain rule because we use it to take derivatives of composties of functions by chaining together their derivatives.[8]
  21. The chain rule can be thought of as taking the derivative of the outer function (applied to the inner function) and multiplying it times the derivative of the inner function.[8]
  22. It is commonly where most students tend to make mistakes, by forgetting to apply the chain rule when it needs to be applied, or by applying it improperly.[8]
  23. The chain rule can be extended to composites of more than two functions.[8]
  24. If the expression is simplified first, the chain rule is not needed.[9]
  25. The chain rule allows the differentiation of composite functions, notated by.[9]
  26. The chain rule is used to differentiate composite functions.[10]
  27. It is often useful to create a visual representation of Equation 4.29 for the chain rule.[11]
  28. The answer is given by the Chain Rule.[12]
  29. However, for our purposes, a chain rule inequality is satisfied 'on average' and this is enough to achieve our goals.[13]
  30. Consequently, the differential operators for which this chain rule holds will be generated by these rotations.[13]
  31. The chain rule could still be used in the proof of this ‘sine rule’.[14]
  32. In any case, the chain rule is not directly needed when working out specific derivatives.[14]
  33. But this fact requires proof; it is the chain rule (or at least a prerequisite for using (1) as the chain rule), and it is not a triviality.[14]
  34. The chain rule can also be discussed as a piece of formal algebra of power series (over a general commutative ring A A ).[14]
  35. Finally, here is a trigonometric function which combines the two previous examples; for this, we will need to repeat the chain rule.[15]
  36. The Chain Rule is a mathematical method to differentiate a composition of functions.[16]
  37. The Chain Rule can be used to differentiate many types of functions.[16]
  38. The Chain Rule can also help us deduce rates of change in the real world.[16]
  39. More importantly for economic theory, the chain rule allows us to find the derivatives of expressions involving arbitrary functions of functions.[17]
  40. Thus the chain rule implies the expression for F'(t) in the result.[17]
  41. Here is a precise result, discovered by Gottfried Wilhelm von Leibniz (1646–1716).As with other expressions obtained by the chain rule, we can interpret each of its parts.[17]
  42. Note: In the Chain Rule, we work from the outside to the inside.[18]
  43. This tutorial presents the chain rule and a specialized version called the generalized power rule.[18]
  44. the question was changed from xto xThe exponential rule is a special case of the chain rule.[18]
  45. In some cases, applying this rule makes differentiation simpler, but this is hardly the power of the chain rule.[19]
  46. Rather, the chain rule is extremely powerful when we do not know what , and/or are.[19]
  47. The chain rule also tells us something about the meaning of the gradient.[19]
  48. Comparing this to the chain rule we see: this tells us that the gradient is orthogonal to the tangent vectors of our level curve.[19]
  49. The chain rule allows us to deal with this case.[20]
  50. It will take a bit of practice to make the use of the chain rule come naturally, it is more complicated than the earlier differentiation rules we have seen.[20]
  51. Using the chain rule, the power rule, and the product rule it is possible to avoid using the quotient rule entirely.[20]
  52. The method is called the "chain rule" because it can be applied sequentially to as many functions as are nested inside one another.[21]
  53. The chain rule has broad applications in physics, chemistry, and engineering, as well as being used to study related rates in many disciplines.[21]
  54. Because one physical quantity often depends on another, which, in turn depends on others, the chain rule has broad applications in physics.[21]
  55. This section presents examples of the chain rule in kinematics and simple harmonic motion.[21]
  56. To do this we need a chain rule for functions of more than one variable.[22]
  57. We may also extend the chain rule to cases when x and y are functions of two variables rather than one.[22]
  58. We may of course extend the chain rule to functions of n variables each of which is a function of m other variables.[22]

소스

  1. ↑ 1.0 1.1 1.2 1.3 Calculus I
  2. ↑ 2.0 2.1 2.2 2.3 Chain rule
  3. ↑ 3.0 3.1 Chain rule | mathematics
  4. ↑ 14.4 The Chain Rule
  5. ↑ 5.0 5.1 3.5 The Chain Rule
  6. ↑ 6.0 6.1 6.2 6.3 Chain rule (video)
  7. ↑ 7.0 7.1 3.6: The Chain Rule
  8. ↑ 8.0 8.1 8.2 8.3 World Web Math: The Chain Rule
  9. ↑ 9.0 9.1 Chain Rule
  10. ↑ The chain rule
  11. ↑ 4.5 The Chain Rule - Calculus Volume 3
  12. ↑ The Chain Rule
  13. ↑ 13.0 13.1 Example sentences
  14. ↑ 14.0 14.1 14.2 14.3 chain rule in nLab
  15. ↑ The Chain Rule
  16. ↑ 16.0 16.1 16.2 The Chain Rule
  17. ↑ 17.0 17.1 17.2 Mathematical methods for economic theory: 2.2 The chain rule
  18. ↑ 18.0 18.1 18.2 Chain Rule (examples, solutions, videos)
  19. ↑ 19.0 19.1 19.2 19.3 The chain rule
  20. ↑ 20.0 20.1 20.2 The chain rule
  21. ↑ 21.0 21.1 21.2 21.3 Wikibooks, open books for an open world
  22. ↑ 22.0 22.1 22.2 The Chain Rule for Functions of Two Variables

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