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PPT On Differentiation

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Published in: Mathematics
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Differentiation is vall about measuring change! Measuring change in a linear function.

Aswathi K / Abu Dhabi

4 years of teaching experience

Qualification: Bachelor of Engineering in AE&I

Teaches: Electronics, Physics, Engineering, Electrical Technology, Biology, Social Studies, Maths, Chemistry, Mathematics

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  1. Differentiation 1 .Rules of Differentiation 2.Applications
  2. Differentiation is all about measuring change! Measuring change in a linear function: Y = a + bx a = intercept b = constant slope i.e. the impact of a unit change in x on the level of y b = Ay Ax 2
  3. If the function is non-linear: e.g. if y = Zü-z 3
  4. The slope of a curve is equal to the slope of the line (or tangent) that touches the curve at that point which is different for different values of x
  5. Example:A firms cost function is 16 Y+AY = (X+AX) Y+AY =x2+2x.Ax+Ax2 AY = x2+2x.Ax+Ax2 - Y since Y = X2 AY = 2X.AX+AX2 = 2X+AX The slope depends on X and AX
  6. The slope of the graph of a function is called the derivative of the function fQO= • The process of differentiation involves letting the change in x become arbitrarily small, i.e. letting A x 0 e.g if = 2X+AX and AX +0 V = 2X in the limit as AX +0
  7. the slope of the non-linear function Y = is 2X • the slope tells us the change in y that results from a very small change in X • We see the slope varies with X e.g. the curve at X = 2 has a slope = 4 and the curve at X = 4 has a slope = 8 • In this example, the slope is steeper at higher values of X
  8. Rules for Differentiation (section 4.3) 1. The Constant Rule If y = c where c is a constant, e.g. Y = 10 dy=o then dx 8
  9. 2. The Linear Function Rule If y = a + bx dy dx dy=6 .. = 10+6x then dx
  10. 3. The Power Function Rule If y = axn, where a and n are constants = n.a.x dy i) y = 4x dy = —8x dy ii) y = 4x2 iii) y = 4x- = 8x —3 10
  11. 4. The Sum-Difference Rule If y = f(x) ± g(x) dx dx dx If y is the sum/difference of two or more functions of x: differentiate the 2 (or more) terms separately, then add/subtract (ii) Y = dy 2x2 + 3x then dx 5x + 4 then 11
  12. 5. The Product Rule 12
  13. Examples ii) 13
  14. 6. The Quotient Rule • If y = u/v where u and v are functions of x ) Then dzz d • 14
  15. Example 1 15
  16. 7. The Chain Rule (Implicit Function Rule) • If y is a function of v, and v is a function of x, then y is a function of x and 16
  17. Examples i) y = (ax2 + bx) — (ax2 + bx) , so y = v let v — 1 1 ax2 + bx å.(2ax + b) 2 = (4x3 + 3x — 7 ) 4 ii) Y — (4x3 + 3x —7 ), so y = v 4 let v — 4 x 3 + x — 7 . 12 x 2 + 3 17
  18. 8. The Inverse Function Rule dy 1 If x = f(y) then dx — dx Examples x = 3y2 then i) = 6y so dx Y = 4x3 then ii) dy = 12x 2 so dy 1 1 2 12x 18
  19. Differentiation in Economics Application I • Total Costs = TC = FC + VC • Total Revenue = TR = Q = Profit = TR - TC • Break even: 0, or TR - • Profit Maximisation: MR = MC 19
  20. Application I: Marginal Functions (Revenue, Costs and Profit) Calculating Marginal Functions c(Tö lwe 20
  21. Example 1 A firm faces the demand curve P=17- • (i) Find an expression for TR in terms of Q • (ii) Find an expression for MR in terms of Q Solution: TR = P.Q= 17Q - 3Q2
  22. Example 2 A firms total cost curve is given by TC=Q3- 4Q2+12Q (i) Find an expression for AC in terms of Q (ii) Find an expression for MC in terms of Q (iii) When does AC=MC? (iv) When does the slope of AC=O? (v) Plot MC and AC curves and comment on the economic significance of their relationship 22
  23. Solution (i) TC = -4Q2 + 12Q Q = Q2—4Q+ 12 Then, AC - d(TC) — 3Q2 —8Q+12 (ii) MC = dQ (iii) When does AC = MC? Q 2 — + 12 — 3Q2- 8Q+ 12 Thus, AC = MC when Q = 2 23
  24. Solution continued.... (iv) When does the slope of AC = 0? d(AC) 2Q — O dQ Q = 2 when slope AC = 0 (v) Economic Significance? MC cuts AC curve at minimum point... 24
  25. 9. Differentiating Exponential Functions If y = exp(x) = ex where e = 2.71828.... dy then dx More generally, If y = Aerx dy = rAe = ry then dx 25
  26. Examples dy l) y = e2x then — = 2e2x dx then 26
  27. 10. Differentiating Natural Logs Recall ify=ex thenx= loæy=lny then dy=ex •Ify=ex dx • nom The Inverse hurtion dx 1 dy Y •Now, if y = ex this is equivalent to writing x = Iny •Thus, x = 1 Y 27
  28. More generally, dy 1 if y = Inx— dx x NOTE: the derivative of a natural log function does not depend on the co-efficient of x dy 1 Thus, if y = Inmx dx x 28
  29. Proof • if y = In mx • Rules of Logs y = Inm+ In x • Differentiating (Sum-Difference rule) dy 1 x 1 x 29
  30. Examples dy _ 1 1) y = In 5x (x>0) dx x 2) y = In(x2+2x+1) let v = (x2+2X+1) so y = In v dy _ dydv Chain Rule: -4 dx dv dx dy dy CIX 1 .(2x + 2) 30
  31. 3) y = x41nx Product Rule: dy 4 3 = x — + Inx. 4x x = x 3 +4x3 Inx = x3(1+41nx) Simplify first using rules of logs y = Inx3 + In(x+2)4 y = 31nx + 41n(x+2) dy 3 4 31
  32. Applications Il how does demand change with a change in price... ... 32
  33. Point elasticity of demand dQP ed = dP'Q ed is negative for a downward sloping demand curve —Inelastic demand if I ed —Unit elastic demand if I ed —Elastic demand if I ed 33
  34. Example 1 -b Find ed of the function Q= ap dQP ed - dÄQ ed = —baP -b -bamb P ecl at all price levels is —b 34
  35. Example 2 If the (inverse) Demand equation is P = 200 — 401n(Q+1) Calculate the price elasticity of demand when Q = 20 dQ p • Price elasticity of demand: ed — dFiQ • P is expressed in terms of Q, CIP 40 dQ Q + 1 clQ Q +1 • Inverse rule 40 0+1 P • Hence, ed Q is 2(_) ed (where P = 200 40 • Q 21 78.22 = -2.05 40• 20 — 401n(20+1) = 78.22) 35
  36. Application Ill: Differentiation of Natural Logs to find Proportional Changes The derivative of log(f(x)) = f(x), or the proportional change in the variable x i.e. y = f(x), then the proportional A x dy 1 — dx Y d(ln y ) Take logs and differentiate to find proportional changes in variables 36
  37. dy 1 1) Show that if y = x", then x and this derivative of In(y) with respect to x. Solution: dy 1 1 clx Y Y 1 y y x x 37
  38. Solution Continued... Now In y = In x u Re-writing In y = alnx d(ln y ) Differentiating the In y with respect to x gives the proportional change in x. 38
  39. Example 2: If Price level at time t is Solution: P(t) = a+bt+ct2 Calculate the rate of inflation. Alternatively, The inflation rate at t is the proportional differentiating the log of P(t) wrttdirectly change in p lnP(t) = In(a+bt+ct2) I dP(t) b + 2Ct where v = (a+bt+Ct2) so Inp = In v 2 Using chain rule, b + 2ct 2 dt 39

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