1.8 Higher Order Derivatives

Math 135 Class Notes

1.8

Business Calculus

Spring 2009

Higher-Order Derivatives

Given a function y = f (x), its derivative f 0 (x) is itself a function. This means we can take its derivative. The derivative of the derivative is called the second derivative of the function and is denoted d2 y f 00 (x) in Newton’s notation or in Leibniz’ notation dx2 We can then take the derivative of the second derivative to obtain the third derivative, denoted d3 y in Leibniz’ notation dx3 We can keep taking derivatives to obtain the fourth derivative, the fifth derivative, and so on. Past the third derivative, we usually use a superscript in parentheses instead of listing primes. f 000 (x) in Newton’s notation

or

d4 y dx4 d5 y f (5) (x) = 5 dx .. . f (4) (x) =

EXAMPLE

Let f (x) = x4 + 2x3 . Find f 0 (x), f 00 (x), f 000 (x), and f (4) (x).

EXAMPLE

Let y = 1/x. Find dy/dx, d2 y/dx2 , and d3 y/dx3 .

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Chapter 1

Differentiation

VELOCITY AND ACCELERATION Suppose the function s(t) represents the distance traveled by an object as a function of time t. Then the derivative s0 (t) is the instantaneous rate of change of the distance, which is the velocity of the object as a function of time. The rate of change of the velocity with respect to time is called the acceleration. The acceleration measures how the velocity is changing with time, so it measures how fast the object is speeding up (for positive acceleration) or slowing down (for negative acceleration). Since the instantaneous acceleration is the derivative of the velocity, which is the derivative of the position function s(t), then the acceleration is the second derivative of the position function. DEFINITION OF VELOCITY AND ACCELERATION The velocity of a moving object whose position at time t is s(t) is given by Velocity = v(t) = s0 (t) =

ds . dt

The acceleration is given by Acceleration = a(t) = v 0 (t) = s00 (t). EXAMPLE When an object is dropped, the distance it falls in t seconds, assuming that air resistance is negligible, is given by s(t) = 16t2 , where s(t) is in feet. If a stone is dropped from a cliff, find each of the following. a) How far it has traveled 5 seconds after being dropped.

b) How fast it is traveling 5 seconds after being dropped.

c) The stone’s acceleration after it has been falling for 5 seconds.