   Chapter 12.2, Problem 39E

Chapter
Section
Textbook Problem

# Using Properties of the Derivative In Exercises 35 and 36, use the properties of the derivative to find the following.(a) r ' ( t ) (b) d d t [ 3 r( t )-u( t ) ] (c) d d t [ (5 t )u( t ) ] (d) d d t [ r ( t ) ⋅ u( t ) ] (e) d d t [ r ( t ) × u( t ) ] (f) d d t [ r ( 2 t ) ] r ( t ) = t i + 3 t j + t 2 k , u ( t ) = 4 t i + t 2 j + t 3 k

(a)

To determine

To calculate: The derivative r'(t) of the function r(t)=ti+3tj+t2k.

Explanation

Given:

The functions are r(t)=ti+3tj+t2k and u(t)=4ti+t2j+t3k.

Formula used:

The general form for differentiation of the expression:

ddt(tn)=ntn1

Calculation:

Consider the following vector valued function:

r(t)=ti+3tj+t2k

Differentiate the function r(t) with respect to t and get the following result:

r

(b)

To determine

To calculate: The expression ddt[3r(t)u(t)] for the functions r(t)=ti+3tj+t2k and u(t)=4ti+t2j+t3k.

(c)

To determine

To calculate: The expression ddt[(5t) u (t)] for the function u(t)=4ti+t2j+t3k

(d)

To determine

To calculate: The expression ddt[r(t)u(t)] for the functions r(t)=ti+3tj+t2k and u(t)=4ti+t2j+t3k.

(e)

To determine

To calculate: The expression ddt[r(t)×u(t)] for the functions r(t)=ti+3tj+t2k and u(t)=4ti+t2j+t3k.

(f)

To determine

To calculate: The expression ddt[r(2t)] for the function r(t)=ti+3tj+t2k.

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