Question

Find the derivative of the vector function.
a)r(t) =<e^(-t),t-t^3,ln t>
b)r(t) =(1/(1 +t)i+(t/(1 +t))j+((t^2)/(1 +t))k

Answer 1

a)

$$r'(t)=\langle -e^{-t}, 1-3t^2, {1 \over t} \rangle$$

b)

$$({t \over 1+t})'={1+t-t \over (1+t)^2}={1 \over (1+t)^2}$$

$$({t^2 \over 1+t})'={2t+2t^2-t^2 \over (1+t)^2}={t^2+2t \over (1+t)^2}$$

$$r'(t)=-{1 \over (1+t)^2}i+{1 \over (1+t)^2}j+{t^2+2t \over (1+t)^2}k$$