Derivatives of Trigonometric Functions, Part 2
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The document discusses finding derivatives of trigonometric functions. It shows that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x). It then uses these results and the chain rule to derive the derivatives of other trig functions like sin^2(x) and tan(x).
Derivatives of Trigonometric Functions, Part 2
- 3. Derivatives of Trigonometric Functions
- 4. Derivatives of Trigonometric Functions In part 1, we showed that:
- 5. Derivatives of Trigonometric Functions In part 1, we showed that: d dx (sin x) = cos x
- 6. Derivatives of Trigonometric Functions In part 1, we showed that: d dx (sin x) = cos x Let’s now find the derivative of f (x) = cos x.
- 7. Derivatives of Trigonometric Functions In part 1, we showed that: d dx (sin x) = cos x Let’s now find the derivative of f (x) = cos x. There are two ways to proceed:
- 8. Derivatives of Trigonometric Functions In part 1, we showed that: d dx (sin x) = cos x Let’s now find the derivative of f (x) = cos x. There are two ways to proceed: 1. Apply the definition of the derivative.
- 9. Derivatives of Trigonometric Functions In part 1, we showed that: d dx (sin x) = cos x Let’s now find the derivative of f (x) = cos x. There are two ways to proceed: 1. Apply the definition of the derivative. 2. Apply the chain rule.
- 10. Derivatives of Trigonometric Functions In part 1, we showed that: d dx (sin x) = cos x Let’s now find the derivative of f (x) = cos x. There are two ways to proceed: 1. Apply the definition of the derivative. 2. Apply the chain rule.
- 11. Derivatives of Trigonometric Functions So, let’s consider the identity:
- 12. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x
- 13. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x We can apply the chain rule directly:
- 14. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x We can apply the chain rule directly: f (x) = 1 2 .
- 15. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x We can apply the chain rule directly: f (x) = 1 2 . 1 1 − sin2 x .
- 16. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x We can apply the chain rule directly: f (x) = 1 2 . 1 1 − sin2 x . (−2 sin x) .
- 17. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x We can apply the chain rule directly: f (x) = 1 2 . 1 1 − sin2 x . (−2 sin x) . cos x
- 18. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x We can apply the chain rule directly: f (x) = 1 2 . 1 1 − sin2 x . (−2 sin x) . cos x = − 2 sin x cos x 2 1 − sin2 x
- 19. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x We can apply the chain rule directly: f (x) = 1 2 . 1 1 − sin2 x . (−2 sin x) . cos x = − ¡2 sin x cos x ¡2 1 − sin2 x
- 20. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x We can apply the chain rule directly: f (x) = 1 2 . 1 1 − sin2 x . (−2 sin x) . cos x = − ¡2 sin x cos x ¡2$$$$$$$Xcos x 1 − sin2 x
- 21. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x We can apply the chain rule directly: f (x) = 1 2 . 1 1 − sin2 x . (−2 sin x) . cos x = − ¡2 sin x cos x ¡2$$$$$$$Xcos x 1 − sin2 x = − sin x cos x cos x =
- 22. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x We can apply the chain rule directly: f (x) = 1 2 . 1 1 − sin2 x . (−2 sin x) . cos x = − ¡2 sin x cos x ¡2$$$$$$$Xcos x 1 − sin2 x = − sin x$$$cos x $$$cos x =
- 23. Derivatives of Trigonometric Functions So, let’s consider the identity: f (x) = cos x = 1 − sin2 x We can apply the chain rule directly: f (x) = 1 2 . 1 1 − sin2 x . (−2 sin x) . cos x = − ¡2 sin x cos x ¡2$$$$$$$Xcos x 1 − sin2 x = − sin x$$$cos x $$$cos x = − sin x = − sin x$$$cos x $$$cos x = − sin x
- 24. Derivatives of Trigonometric Functions So, we have that:
- 25. Derivatives of Trigonometric Functions So, we have that: d dx (cos x) = − sin x
- 26. Derivatives of Trigonometric Functions So, we have that: d dx (cos x) = − sin x
- 27. Derivatives of Trigonometric Functions So, we have that: d dx (cos x) = − sin x We can now solve some problems.
- 28. Example 1 Let’s find the derivative of the function:
- 29. Example 1 Let’s find the derivative of the function: f (x) = sin2 x
- 30. Example 1 Let’s find the derivative of the function: f (x) = sin2 x We’ve already found this derivative when we calculated the derivative of cos x.
- 31. Example 1 Let’s find the derivative of the function: f (x) = sin2 x We’ve already found this derivative when we calculated the derivative of cos x. We apply the chain rule:
- 32. Example 1 Let’s find the derivative of the function: f (x) = sin2 x We’ve already found this derivative when we calculated the derivative of cos x. We apply the chain rule: f (x) = 2 sin x d dx (sin x)
- 33. Example 1 Let’s find the derivative of the function: f (x) = sin2 x We’ve already found this derivative when we calculated the derivative of cos x. We apply the chain rule: f (x) = 2 sin x d dx (sin x) = 2 sin x cos x
- 34. Example 1 Let’s find the derivative of the function: f (x) = sin2 x We’ve already found this derivative when we calculated the derivative of cos x. We apply the chain rule: f (x) = 2 sin x d dx (sin x) = 2 sin x cos x
- 35. Example 2
- 36. Example 2 f (x) = tan x
- 37. Example 2 f (x) = tan x Here we can use the product rule:
- 38. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x =
- 39. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1
- 40. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) =
- 41. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x.
- 42. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x. (cos x)−1 +
- 43. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x. (cos x)−1 + d dx (sin x) . (cos x)−1
- 44. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x. (cos x)−1 + d dx (sin x) . (cos x)−1 = sin x.
- 45. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x. (cos x)−1 + d dx (sin x) . (cos x)−1 = sin x.(−1). (cos x)−2 . d dx (cos x) +
- 46. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x. (cos x)−1 + d dx (sin x) . (cos x)−1 = sin x.(−1). (cos x)−2 . d dx (cos x) + cos x. (cos x)−1
- 47. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x. (cos x)−1 + d dx (sin x) . (cos x)−1 = sin x.(−1). (cos x)−2 . ¨¨ ¨¨¨¨B− sin x d dx (cos x) + cos x. (cos x)−1
- 48. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x. (cos x)−1 + d dx (sin x) . (cos x)−1 = sin x.(−1). (cos x)−2 . ¨¨¨ ¨¨¨B− sin x d dx (cos x) + cos x. (cos x)−1 = sin2 x. cos2 x +
- 49. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x. (cos x)−1 + d dx (sin x) . (cos x)−1 = sin x.(−1). (cos x)−2 . ¨¨¨ ¨¨¨B− sin x d dx (cos x) + cos x. (cos x)−1 = sin2 x. cos2 x + cos x cos x
- 50. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x. (cos x)−1 + d dx (sin x) . (cos x)−1 = sin x.(−1). (cos x)−2 . ¨¨¨ ¨¨¨B− sin x d dx (cos x) + cos x. (cos x)−1 = sin2 x. cos2 x + & & &b 1 cos x cos x
- 51. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x. (cos x)−1 + d dx (sin x) . (cos x)−1 = sin x.(−1). (cos x)−2 . ¨¨¨ ¨¨¨B− sin x d dx (cos x) + cos x. (cos x)−1 = sin2 x. cos2 x + & & &b 1 cos x cos x = tan2 x + 1
- 52. Example 2 f (x) = tan x Here we can use the product rule: f (x) = sin x cos x = f (x) = sin x cos x = sin x. (cos x)−1 f (x) = sin x. (cos x)−1 + d dx (sin x) . (cos x)−1 = sin x.(−1). (cos x)−2 . ¨¨¨ ¨¨¨B− sin x d dx (cos x) + cos x. (cos x)−1 = sin2 x. cos2 x + & & &b 1 cos x cos x = tan2 x + 1