![𝑑2𝑦
𝑑𝑥2]
න
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∞
𝑥𝑛
න
0
∞
𝑥𝑛
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𝑥𝑛
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Differentiation
Properties of derivatives
Applications of derivatives](https://image.slidesharecdn.com/differentiationanditsproperties3rdaug-241225171921-edb21c7b/85/Differentiation-and-it-s-properties-3rd-AUg-pdf-4-320.jpg)
![𝑑2𝑦
𝑑𝑥2]
න
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𝑥𝑛
න
0
∞
𝑥𝑛
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𝑥𝑛
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Differentiation
𝑑2𝑦
𝑑𝑥2]
න
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∞
𝑥𝑛
න
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∞
𝑥𝑛
න
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𝑥𝑛
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𝑥𝑛](https://image.slidesharecdn.com/differentiationanditsproperties3rdaug-241225171921-edb21c7b/85/Differentiation-and-it-s-properties-3rd-AUg-pdf-5-320.jpg)
![𝑑2𝑦
𝑑𝑥2]
න
0
∞
𝑥𝑛
න
0
∞
𝑥𝑛
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𝑥𝑛
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𝑥𝑛
න
0
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𝑥𝑛
Differentiation
Properties of derivatives](https://image.slidesharecdn.com/differentiationanditsproperties3rdaug-241225171921-edb21c7b/85/Differentiation-and-it-s-properties-3rd-AUg-pdf-14-320.jpg)
![Chain rule:
𝑑𝑓
𝑑𝑥
=
𝑑𝑓
𝑑𝑔
.
𝑑𝑔
𝑑𝑥
Differentiate
outer function
Differentiate
inner function
If 𝑓 = 𝑓 (𝑔); 𝑔 = 𝑔 (𝑥)
𝑓’ (𝑥) = 𝑓’ ( 𝑔(𝑥) ) 𝑔’ (𝑥)
Example:
𝑓 𝑥 = sin(𝑥)
𝑔 𝑥 = 𝑥2
𝑓[𝑔 𝑥 ] = sin(𝑥2)](https://image.slidesharecdn.com/differentiationanditsproperties3rdaug-241225171921-edb21c7b/85/Differentiation-and-it-s-properties-3rd-AUg-pdf-26-320.jpg)
![𝑑2𝑦
𝑑𝑥2]
න
0
∞
𝑥𝑛
න
0
∞
𝑥𝑛
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𝑥𝑛
න
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𝑥𝑛
Differentiation
Properties of derivatives
Applications of derivatives](https://image.slidesharecdn.com/differentiationanditsproperties3rdaug-241225171921-edb21c7b/85/Differentiation-and-it-s-properties-3rd-AUg-pdf-32-320.jpg)
Differentiation and it's properties 3rd AUg.pdf
- 4. 𝑑2𝑦 𝑑𝑥2] න 0 ∞ 𝑥𝑛 න 0 ∞ 𝑥𝑛 න 0 ∞ 𝑥𝑛 න 0 ∞ 𝑥𝑛 න 0 ∞ 𝑥𝑛 Differentiation Properties of derivatives Applications of derivatives
- 6. To study the variation of one quantity with respect to another quantity. To study slope of a curve. To find the maximum and minimum values of any function. Differentiation is all about finding rate of change of one quantity with respect to another.
- 7. Speed = distance time Acceleration = change in velocity time Power = work done time Counts per time ( eg. Heart rate, radioactive decay rate ) Speed = no.of counts time 𝑣 𝑢 𝑡 𝑎
- 8. Tangent: Secant: Secant is a line which intersects a curve at two different points 𝑦 𝑥 𝐴 Δ𝑥 Δ𝑦 𝐵 𝑥 𝑦 𝐴 Δ𝑥 Δ𝑦 𝐵 𝑦 𝑥 𝐴 Δ𝑥 Δy 𝐵 𝑦 𝑥 Tangent Secant A line which touches a curve at one point A limiting case of secant which intersects the curve at two infinitesimally close points.
- 9. Tangent is the limiting case of secant: Tangent Secant P 𝑥 𝑦 ((𝑥, 𝑓(𝑥)) Δ𝑥 Q R S (𝑥 + ∆𝑥, 𝑓(𝑥 + ∆𝑥)) Δ𝑦 For secant 𝑃𝑄, Slope = ∆𝑦 ∆𝑥 For tangent at 𝑃, Slope = 𝑙𝑖𝑚 ∆𝑥→0 ∆𝑦 ∆𝑥 = 𝑑𝑦 𝑑𝑥 = 𝑦′ = 𝑓′(𝑥) ℎ ∆𝑥 ∆𝑦 Slope 0 𝑑𝑥 𝑑𝑦 𝑑𝑦 𝑑𝑥 Derivative of a function at a point gives slope of tangent at that point.
- 10. Derivative of a function: Slope = ∆𝑦 ∆𝑥 Slope = rate of change = 𝑙𝑖𝑚 ∆𝑥→0 ∆𝑦 ∆𝑥 𝑓 (𝑥) 𝑑 𝑑𝑥 𝑓 ′ (𝑥)
- 11. 𝑑 𝑑𝑥 (𝑥𝑛 ) = 𝑛𝑥𝑛−1 𝑑𝑦 𝑑𝑥 = 𝑦′ = 𝑓′(𝑥) 𝑓(𝑥) = 𝑥𝑛 Power rule of differentiation:
- 12. Differentiation of trigonometric functions: (1) 𝑑 𝑑𝑥 sin 𝑥 = cos 𝑥 (2) 𝑑 𝑑𝑥 cos 𝑥 = − sin 𝑥 (3) 𝑑 𝑑𝑥 tan 𝑥 = sec2 𝑥 (4) 𝑑 𝑑𝑥 cot 𝑥 = − cosec2 𝑥 (5) 𝑑 𝑑𝑥 sec 𝑥 = sec 𝑥 tan 𝑥 (6) 𝑑 𝑑𝑥 cosec 𝑥 = − cosec 𝑥 cot 𝑥
- 13. Derivative of logarithmic function: 𝑑 𝑑𝑥 (𝑒𝑥) = 𝑒𝑥 𝑑 𝑑𝑥 (𝑎𝑥) = 𝑎𝑥 ln (𝑎) 𝑑 𝑑𝑥 (ln 𝑥) = 1 𝑥 "its slope is its value" 𝑦 = 𝑓(𝑥) = 𝑒𝑥 𝑥 𝑦 Derivative of exponential function:
- 15. Derivative of constant times a function: 𝑑 𝑑𝑥 𝑎𝑓 𝑥 = 𝑎 𝑑𝑓 𝑥 𝑑𝑥
- 16. 𝑑 𝑑𝑥 𝑓 𝑥 ± 𝑔(𝑥) = 𝑑𝑓 𝑥 𝑑𝑥 ± 𝑑𝑔 𝑥 𝑑𝑥 Derivative of sum or difference of two functions:
- 17. Product rule: 𝑑 𝑓 𝑥 . 𝑔(𝑥) 𝑑𝑥 = 𝑑𝑓 𝑥 𝑑𝑥 𝑔 𝑥 + 𝑑𝑔 𝑥 𝑑𝑥 𝑓 𝑥
- 18. a) 𝑥 cos 𝑥 + 2𝑥 sin 𝑥 b) 𝑥2 cos 𝑥 + 𝑥2 sin 𝑥 c) 𝑥2 cos 𝑥 + 2𝑥 sin 𝑥 d) 𝑥2 cos 𝑥 − 2𝑥 sin 𝑥 For 𝑓(𝑥) = 𝑥2 cos 𝑥 ; 𝑑 𝑑𝑥 𝑓(𝑥) =? Exercise
- 19. Applying, 𝑑 𝑑𝑥 𝑢 𝑣 = 𝑢 𝑑𝑣 𝑑𝑥 + 𝑣 𝑑𝑢 𝑑𝑥 𝑑 𝑑𝑥 𝑓(𝑥) = −𝑥2 sin 𝑥 + 2𝑥 cos 𝑥 For 𝑓(𝑥) = 𝑥2 cos 𝑥 ; 𝑑 𝑑𝑥 𝑓(𝑥) =? Solution
- 20. a) 𝑥 cos 𝑥 + 2𝑥 sin 𝑥 b) 𝑥2 cos 𝑥 + 𝑥2 sin 𝑥 c) −𝑥2 sin 𝑥 + 2𝑥 co𝑠 𝑥 d) 𝑥2 cos 𝑥 − 2𝑥 sin 𝑥 For 𝑓(𝑥) = 𝑥2 cos 𝑥 ; 𝑑 𝑑𝑥 𝑓(𝑥) =? Answer
- 21. Division rule: 𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 = 𝑑𝑓 𝑥 𝑑𝑥 𝑔 𝑥 − 𝑑𝑔 𝑥 𝑑𝑥 𝑓 𝑥 (𝑔 𝑥 )2
- 22. a) cos 𝑥−𝑥 sin 𝑥 cos2𝑥 b) cos 𝑥+𝑥 sin 𝑥 cos2𝑥 c) cos 𝑥+𝑥 sin 𝑥 cos𝑥 d) cos 𝑥+𝑥2 sin 𝑥 cos2𝑥 For 𝑓(𝑥) = 𝑥 cos 𝑥 ; 𝑑 𝑑𝑥 𝑓(𝑥) =? Exercise
- 23. 𝑑 𝑑𝑥 𝑢 𝑣 = 𝑣 𝑑𝑢 𝑑𝑥 − 𝑢 𝑑𝑣 𝑑𝑥 𝑣2 𝑑 𝑑𝑥 𝑓(𝑥) = cos 𝑥 + 𝑥 sin 𝑥 cos2𝑥 𝑑 𝑑𝑥 𝑥 cos 𝑥 = cos x d dx x − x d dx cos x cos2x For 𝑓(𝑥) = 𝑥 cos 𝑥 ; 𝑑 𝑑𝑥 𝑓(𝑥) =? Solution
- 24. a) cos 𝑥−𝑥 sin 𝑥 cos2𝑥 b) cos 𝑥+𝑥 sin 𝑥 cos2𝑥 c) cos 𝑥+𝑥 sin 𝑥 cos𝑥 d) cos 𝑥+𝑥2 sin 𝑥 cos2𝑥 For 𝑓(𝑥) = 𝑥 cos 𝑥 ; 𝑑 𝑑𝑥 𝑓(𝑥) =? Answer
- 25. Derivative of constant times a function: 𝑑 𝑑𝑥 𝑓 𝑥 ± 𝑔(𝑥) = 𝑑𝑓 𝑥 𝑑𝑥 ± 𝑑𝑔 𝑥 𝑑𝑥 𝑑 𝑑𝑥 𝑎𝑓 𝑥 = 𝑎 𝑑𝑓 𝑥 𝑑𝑥 Derivative of sum or difference of two functions: Product rule: 𝑑 𝑓 𝑥 . 𝑔(𝑥) 𝑑𝑥 = 𝑑𝑓 𝑥 𝑑𝑥 𝑔 𝑥 + 𝑑𝑔 𝑥 𝑑𝑥 𝑓 𝑥 Division rule: 𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 = 𝑑𝑓 𝑥 𝑑𝑥 𝑔 𝑥 − 𝑑𝑔 𝑥 𝑑𝑥 𝑓 𝑥 (𝑔 𝑥 )2
- 26. Chain rule: 𝑑𝑓 𝑑𝑥 = 𝑑𝑓 𝑑𝑔 . 𝑑𝑔 𝑑𝑥 Differentiate outer function Differentiate inner function If 𝑓 = 𝑓 (𝑔); 𝑔 = 𝑔 (𝑥) 𝑓’ (𝑥) = 𝑓’ ( 𝑔(𝑥) ) 𝑔’ (𝑥) Example: 𝑓 𝑥 = sin(𝑥) 𝑔 𝑥 = 𝑥2 𝑓[𝑔 𝑥 ] = sin(𝑥2)
- 27. Exercise a) 10𝑒10𝑥 b) 𝑒4𝑥 c) 𝑒4𝑥 4 d) 𝑒16𝑥 Differentiate 𝑓(𝑥) = 𝑒10𝑥
- 28. 𝑑 𝑑𝑥 𝑒10𝑥 = 10𝑒10𝑥 Differentiate 𝑓(𝑥) = 𝑒10𝑥 Solution
- 29. a) 10𝑒10𝑥 b) 𝑒4𝑥 c) 𝑒4𝑥 4 d) 𝑒16𝑥 Differentiate 𝑓(𝑥) = 𝑒10𝑥 Answer
- 32. 𝑑2𝑦 𝑑𝑥2] න 0 ∞ 𝑥𝑛 න 0 ∞ 𝑥𝑛 න 0 ∞ 𝑥𝑛 න 0 ∞ 𝑥𝑛 න 0 ∞ 𝑥𝑛 Differentiation Properties of derivatives Applications of derivatives
- 33. 𝑓 (𝑥) f’(x) 𝑑 𝑑𝑥 𝑓’’(𝑥) 𝑑 𝑑𝑥 𝑓 ′ (𝑥) 𝑑2 𝑦 𝑑𝑥2 = 𝑑 𝑑𝑥 𝑑𝑦 𝑑𝑥 Example: 𝑓 𝑥 = sin(𝑥) 𝑓′ 𝑥 = cos(𝑥) 𝑓′′ 𝑥 = − 𝑠𝑖𝑛(𝑥)
- 34. Double-Differentiation of trigonometric functions: (1) (2) 𝑑2 𝑑𝑥2 sin 𝑥 = −sin 𝑥 𝑑2 𝑑𝑥2 cos 𝑥 = − cos 𝑥
- 35. 𝑓(𝑥1) 𝑓(𝑥2) 𝑥2 𝑥1 𝑥 𝑦 Increasing function: Decreasing function: Function is increasing: when 𝑥1 < 𝑥2 𝑓 𝑥1 ≤ 𝑓(𝑥2) and 𝑓(𝑥2) 𝑓(𝑥1) 𝑥2 𝑥1 𝑥 𝑦 Function is decreasing: when 𝑥1 < 𝑥2 and 𝑓 𝑥1 ≥ 𝑓(𝑥2)
- 36. 𝑥 increases Decreasing function 𝑦 decreases 𝑥 increases 𝑦 increases Increasing function 𝑥 𝑦 𝑥1 𝑥2 𝑥3 𝑥4 𝑦1 𝑦2 −𝑦3 −𝑦4 𝑥 𝑦 Tangent has positive slope Tangent has negative slope Increasing Decreasing Increasing 𝑓′(𝑥) > 0 𝑓′(𝑥) < 0 𝑥 𝑦 Tangent has zero slope Decreasing Increasing Increasing 𝑓′ 𝑥 = 0 𝑓′ 𝑥 = 0 (Critical Point) The point at which tangent has zero slope is called critical point.
- 39. 5 3 1 𝑦 𝑥 2 4 For maxima, as 𝑥 increases the slope decreases. Condition for maxima: 𝑑𝑦 𝑑𝑥 = 0 𝑑2𝑦 𝑑𝑥2 < 0 and 5 1 𝑦 𝑥 2 3 4 For minima, as 𝑥 increases the slope increases. Condition for maxima: 𝑑𝑦 𝑑𝑥 = 0 𝑑2𝑦 𝑑𝑥2 > 0 and
- 40. To study the variation of one quantity with respect to another quantity. Differentiation is all about finding rate of change of one quantity with respect to another. To study slope of a curve. To find the maximum and minimum values of any function. Instantaneous rate of change: Rate of change of 𝑦 with respect to 𝑥 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 = Δ𝑦 Δ𝑥 Velocity (𝑣) = rate of change of position(s) with respect to time As 𝛥 → 0, Rate → 𝑑𝑦 𝑑𝑥 𝑣 = 𝑑𝑠 𝑑𝑡
- 41. a) 0.14 𝑚 b) 1.4 𝑚 c) 2.8 𝑚 d) 14 𝑚 A ball is thrown in the air. It’s height at any point is given by ℎ = 3 + 14𝑡 − 5𝑡2. What is maximum height (𝑚) attained by the ball? Exercise
- 42. For a maxima of a given function, 𝑓′ 𝑥 = 0 14 − 10𝑡 = 0 A ball is thrown in the air. It’s height at any point is given by ℎ = 3 + 14𝑡 − 5𝑡2. What is maximum height (𝑚) attained by the ball? ℎ′ = 0 + 14 − 10𝑡 For maximum height, ℎ′= 0 𝑡 = 1.4 𝑚 Solution
- 43. a) 0.14 𝑚 b) 1.4 𝑚 c) 2.8 𝑚 d) 14 𝑚 A ball is thrown in the air. It’s height at any point is given by ℎ = 3 + 14𝑡 − 5𝑡2. What is maximum height (𝑚) attained by the ball? Answer
- 45. a) 3 b) −1 c) 1 d) −3 Find the local minima for the function, 𝑦 = 2𝑥3 − 6𝑥 + 2 Exercise
- 46. 𝑓′ 𝑥 = 0 𝑓"(1) = 12 Find the local minima for the function, 𝑦 = 2𝑥3 − 6𝑥 + 2 6𝑥2 − 6 = 0 𝑥 = ±1 𝑓"(𝑥) = 12𝑥 Given function has a local minima at 𝑥 = 1 Solution
- 47. a) 3 b) −1 c) 1 d) −3 Find the local minima for the function, 𝑦 = 2𝑥3 − 6𝑥 + 2 Answer