Lecture 05 Differentiation 1
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The learning outcomes of this topic are: - Find the derivative of variables raised to a power - Use the rules of differentiation - Relate differentiation to optimization (Obtain the economic order quantity formula) This topic will cover: - Gradient - Definition of the derivative - Rules of differentiation
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Lecture 05 Differentiation 1
- 2. This topic will cover: ◦ Gradient ◦ Definition of the derivative ◦ Rules of differentiation
- 3. By the end of this topic students will be able to: ◦ Find the derivative of variables raised to a power ◦ Use the rules of differentiation ◦ Relate differentiation to optimization Obtain the economic order quantity formula
- 4. ◦ Expression2𝑥, 5𝑥 + 1, 10𝑥2 + 2𝑥 ◦ Equation identity 2𝑥 + 3 5𝑥 + 1 = 10𝑥2 + 17𝑥 + 3 conditional 2 − 𝑥2 = 0, 𝑦 − 𝑥2 = 0 ◦ Function 𝑥 = 𝑦 , 𝑦 ≥ 0 𝑥 = 𝑦 , 𝑦 ≥ 0, 𝑥 ≥ 0
- 5. -4 -2 0 2 0 0.5 1 1.5 2 0 1 2 3 4 0 0.5 1 1.5 2 0 2 4 6 8 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 y = x y = x2 y = ex y = lnx
- 6. y x y = c y = f(x) = mx + c x = -c/m y increases m x increases 1
- 7. ◦ Ratio of vertical change to horizontal change gradient = ∆𝑦 ∆𝑥 ∆𝑦 ∆𝑥 = 𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥 ∆𝑥 ∆𝑦 ∆𝑥 = m𝑥 + m∆𝑥 + c − m𝑥 − c ∆𝑥 ∆𝑦 ∆𝑥 = m f(x+∆x) f(x) x+∆xx ∆x ∆y= f(x+∆x)-f y = f(x) = mx + c
- 8. ◦ Which line best approximates the gradient of the non- linear curve at the point? x xxx
- 9. ◦ Derivative quantifies rate of change of a function with respect to an independent variable aka differential coefficient ◦ Defined as 𝑓′ 𝑥 = 𝑑𝑦 𝑑𝑥 = lim ∆𝑥→0 𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥 ∆𝑥 ∆y= f(x+∆x)-f( f(x+∆x) f(x) x+∆xx ∆x
- 10. 𝑓′ 𝑥 = 𝑑𝑦 𝑑𝑥 = lim ∆𝑥→0 𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥 ∆𝑥 𝑓 𝑥 = 𝑦 = a𝑥2 𝑓′ 𝑥 = 𝑑𝑦 𝑑𝑥 = lim ∆𝑥→0 a 𝑥 + ∆𝑥 2 − 𝑥2 ∆𝑥 ∆y= a(x+∆x)2 – ∆x y y = ax2 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥 = 2ax 𝑑𝑦 𝑑𝑥 = lim ∆𝑥→0 2a𝑥 + a∆𝑥 = 2a𝑥
- 11. ◦ Already shown that for, 𝑦 = a𝑥2 , 𝑑𝑦 𝑑𝑥 = 2a𝑥 ◦ Generally it can also be shown, 𝑦 = a𝑥n, 𝑑𝑦 𝑑𝑥 = na𝑥n−1
- 12. a) 𝑦 = 3𝑥 b) 𝑦 = 4𝑥3 c) 𝑦 = 7 𝑥 d) 𝑦 = 17 e) 𝑦 = 3 𝑥 f) 𝑦 = 6 𝑥 𝒅𝒚 𝒅𝒙 = 𝐧𝐚𝒙 𝒏−𝟏𝒚 = 𝐚𝒙 𝒏
- 13. ◦ Commonly met functions in business powers, exponential and natural logarithms. 𝒚 𝒅𝒚 𝒅𝒙 𝒚 = 𝐚𝒙 𝒏 𝒅𝒚 𝒅𝒙 = 𝐧𝐚𝒙 𝐧−𝟏 𝒚 = 𝐞 𝐚𝒙 𝒅𝒚 𝒅𝒙 = 𝐚𝐞 𝐚𝒙 𝒚 = 𝒍𝒏 𝒂𝒙 = 𝒍𝒐𝒈 𝐞 𝒙 𝒅𝒚 𝒅𝒙 = 𝟏 𝒙
- 14. a) 𝑦 = 𝑒5𝑥 b) 𝑦 = 𝑒−4𝑥 a) 𝑑𝑦 𝑑𝑥 = 5𝑒5𝑥 b) 𝑑𝑦 𝑑𝑥 = −4𝑒−4𝑥 𝒅𝒚 𝒅𝒙 = 𝐚𝒆 𝒂𝒙𝒚 = 𝐞 𝐚𝒙 a) 𝑦 = 𝑙𝑛(5𝑥) b) 𝑦 = 𝑙𝑛 −4𝑥 a) 𝑑𝑦 𝑑𝑥 = 1 𝑥 b) 𝑑𝑦 𝑑𝑥 = 1 𝑥 𝒅𝒚 𝒅𝒙 = 𝟏 𝒙 𝒚 = 𝒍𝒏 𝒂𝒙
- 15. 𝑑 𝑑𝑥 𝑎𝑓 𝑥 + 𝑏𝑔 𝑥 = 𝑎 𝑑𝑓 𝑑𝑥 + 𝑏 𝑑𝑔 𝑑𝑥 𝑦 = 4𝑥3 + 3𝑥 + 17 𝑑𝑦 𝑑𝑥 = 12𝑥2 + 3
- 16. 𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 = 𝑓 𝑥 𝑑𝑔 𝑑𝑥 + 𝑔 𝑥 𝑑𝑓 𝑑𝑥 𝑦 = 𝑥4e2𝑥 𝑑𝑦 𝑑𝑥 = 𝑥4 2e2𝑥 + e2𝑥 4𝑥3 = 2𝑥3 e2𝑥 𝑥 + 2 𝑓(𝑥) = 𝑥4 𝑔(𝑥) = e2𝑥 𝑑𝑓 𝑑𝑥 = 4𝑥3 𝑑𝑔 𝑑𝑥 = 2e2𝑥
- 17. 𝑑 𝑑𝑥 𝑓 𝑔 𝑥 = 𝑑𝑓 𝑑𝑔 𝑑𝑔 𝑑𝑥 𝑦 = 7𝑥2 + 3 3 𝑑𝑓 𝑑𝑔 = 3𝑔2 𝑑𝑔 𝑑𝑥 = 14𝑥 𝑓 = 𝑔(𝑥) 3 𝑑𝑦 𝑑𝑥 = 42𝑥 7𝑥2 + 3 2 𝑔 𝑥 = 7𝑥2 + 3
- 18. 𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 = 𝑔 𝑥 𝑑𝑓 𝑑𝑥 −𝑓 𝑥 𝑑𝑔 𝑑𝑥 𝑔 𝑥 2 𝑦 = 𝑥 𝑥2 + 1 𝑑𝑔 𝑑𝑥 = 2𝑥 𝑑𝑓 𝑑𝑥 = 1 𝑔 𝑥 = 𝑥2 + 1 𝑑𝑦 𝑑𝑥 = 1 − 𝑥2 𝑥2 + 1 2 𝑓 𝑥 = 𝑥
- 19. 𝑑 𝑑𝑥 𝑎𝑓 𝑥 + 𝑏𝑔 𝑥 = 𝑎 𝑑𝑓 𝑑𝑥 + 𝑏 𝑑𝑔 𝑑𝑥 𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 = 𝑓 𝑥 𝑑𝑔 𝑑𝑥 + 𝑔 𝑥 𝑑𝑓 𝑑𝑥 𝑑 𝑑𝑥 𝑓 𝑔 𝑥 = 𝑑𝑓 𝑑𝑔 𝑑𝑔 𝑑𝑥 𝑑 𝑑𝑥 𝑓 𝑥 𝑔 𝑥 = 𝑔 𝑥 𝑑𝑓 𝑑𝑥 −𝑓 𝑥 𝑑𝑔 𝑑𝑥 𝑔 𝑥 2
- 21. ◦ Economic order quantity1 𝑇𝐶 = 𝑑. 𝑝 + 𝐶ℎ 𝑞 2 + 𝐶 𝑜 𝑑 𝑞 1 Harris (1913), Wilson (1934) totalcost quantity 𝑑𝑇𝐶 𝑑𝑞 = 𝐶ℎ 2 − 𝐶 𝑜 𝑑 𝑞2 𝐶ℎ 2 − 𝐶 𝑜 𝑑 𝑞2 = 0 𝑞 = 2𝐶0 𝑑 𝐶ℎ
- 22. By the end of this topic students will be able to: ◦ Find the derivative of variables raised to a power ◦ Use the rules of differentiation ◦ Relate differentiation to optimization Obtain the economic order quantity formula
- 23. ◦ Harris, FW. (1913). How many parts to make at once. Factory, the Magazine of Management 10(2), 135-136; reprinted in Operations Research 1990, 38(6), 947-950 ◦ Wilson, RH. (1934). A scientific routine for stock control, Harvard Business Review 13(1), 116-128.
- 24. Any Questions?