Higherorder Derivatives

The sccond-order derivative, written /"(*), measures the slope and the rate of changc of the first derivative, just as the first denvative measures the slope and the rate of change of the original or primitive function. The third-order derivative f"(x) measures the slope and rate of change of the second-order derivative, etc. Higher-order derivatives arc found by applying the rules of differentiation to lower-order derivatives, as illustrated in Example 14 and treated in Problems 3.20 and 321.

EXAMPLE 14. Given y = f{x). common notation for the second-order derivauve includes f(x). d'y/dx1. y\ and £>*y: for the third-order derivative, diy!dxi, y*. and D'y; for the fourth-order derivative*. f'\x). d*y/dx*. and D*y, etc.

Higher-order derivatives are found by successively applying the rules of differentiation to derivatives of the previous order. Thus, if fix) - Zx* + 5.r + 3 x\

f'(x) - 8r' + 15r + tx f(x) - 24r + 3ftr + 6 Fix) « 48r + 30 f*\x) - 48 /*'»(*) " 0

Sec Problems 3.20 and 3.21.

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