Ex: 9 144Chapter 3: DerivativesHow to Read the Symbols forDerivativesIf y" is differentiable, its derivative, y" = dy"/dx = d'y/dx', is the third derivativeof y with respect to x. The names continue as you imagine, withy'"y prime"yla) - d ya-1)yn-1) -= D'yy""y double prime"dr"d'y"d squared y dx squared"drdenoting the nth derivative of y with respect to x for any positive integer n.We can interpret the second derivative as the rate of change of the slope of the tangentto the graph of y = f(x) at each point. You will see in the next chapter that the secondderivative reveals whether the graph bends upward or downward from the tangent line aswe move off the point of tangency. In the next section, we interpret both the second andthird derivatives in terms of motion along a straight line."y triple prime"yin "y super n"dy* "d to the n of y by dx to the n"dr"The first four derivatives of y= x - 3x2 + 2 arey = 3x - 6xD"D to the n"EXAMPLE 10First derivative:Second derivative: y" = 6x – 6Third derivative:y" = 6Fourth derivative: yld) = 0.All polynomial functions have derivatives of all orders. In this example, the fifth and laterderivatives are all zero.Exercises 3.3Derivative CalculationsIn Exercises 1-12, find the first and second derivatives.1. y - -x + 31 + x - 4Vxva)25. v=26. r= 2Ve+ Ve2. y =x +x + 84. w - 32 - 7 + 212(x+ )(x + 2)(x - 1)(x - 2)3. s- 5 - 327. y=(x - 1) (x + x + 1)28. y=5. y =4x-x + 2e6. y =+5+e29. y= 2e + e30. y=+ 3e2e' - x30. у7. w = 3; -8. s--2 +31. y = x'e33. y- x4 +32. w-re34. y-xS + 29. y = 6r - 10x - 5x10. y = 4 - 2x - x35. s= 212 + 3e36, w -+ T511. r=-12. - 2-1 +!12. r=4z14Vz3s2s37. y- V -38. y- Vx"6 + 2el3In Exercises 13-16, find y' (a) by applying the Product Rule and(b) by multiplying the factors to produce a sum of simpler terms to39. r=+ 2)40, r=differentiate.13. y = (3 - x) (x - x + 1) 14. y = (2r + 3)(5x - 4x)D(x + 5 + !) 16. y = (1 + x*)(x4 – ")Find the derivatives of all orders of the functions in Exercises 41-44.15. y = (x + 1)x16. y = (1 + x')(x4 - x*)41. y =- - x42. y=212043. y = (x - 1)(x + 2)(x + 3) 44. y = (4x? + 3X2 - x)xFind the derivatives of the functions in Exercises 17-40.18. =4- 3r3x + x2x + 5Find the first and second derivatives of the functions in Exercises17. y =18. z=Зх - 245-52.19. glr) = - 4x + 0.545. y = +1? + 5r – 120. f) =+t-246. s=19. g(x)-2 +- 221. v- (1 - n(1 + )-22. w = (2x - 7)x + 5)(0 - )(0 + 0 + 1)(x + xMx - x + I)47. r48. u=23, fr) - V - IVs +1Sx + 123. f(s) -24. -q + 31-1+ (q + 1)I+ 323- 2)49. w-50. р32(9 -

Name: Ex: 9 144Chapter 3: DerivativesHow to Read the Symbols forDerivatives
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Description: Ex: 9 144Chapter 3: DerivativesHow to Read the Symbols forDerivativesIf y" is differentiable, its derivative, y" = dy"/dx = d'y/dx', is the third derivativeof y with respect to x. The names continue as you imagine, withy'"y prime"yla) - d ya-1)yn-1)