Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 14 - Calculus of Vector-Valued Functions - 14.2 Calculus of Vector-Valued Functions - Exercises - Page 720: 18


$$\frac{d}{dt}(t^4r_1(t)) =\langle 6 t^5,7t^6,5t^4 \rangle .$$

Work Step by Step

Since $$ r_1(t)=\langle t^2,t^3,t \rangle $$ then, we have $$\frac{d}{dt}(t^4r_1(t))=\frac{d}{dt}(t^4)r_1(t)) +t^4 \frac{d}{dt}( r_1(t))\\ =4t^3\langle t^2,t^3,t \rangle+t^4 \langle 2t,3t^2,1 \rangle\\ =\langle 6 t^5,7t^6,5t^4 \rangle .$$
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