Find the principal unit normal vector?

Question

Find the principal unit normal vector to the curve defined by r(t)=<t, t^2>.

T(t)=1/sqrt(1+4t^2)i+2t/sqrt(1+4t^2)j

I need to find T'(t) using quotient rule but don't know how. Show your steps.

Answer 1

T  =  T(t)=1/sqrt(1+4t^2)i+2t/sqrt(1+4t^2)j

T'  =  i (-1/2)(8t)(1 + 4t^2)^(-1/2)/(1 + 4t^2)  +  J (2(1 + 4t^2)^(1/2) - (2t)(1/2)(8t)(1 + 4t^2)^(-1/2))/(1 + 4t^2)

 

=  (i  (-4t)/(1 + 4t^2)^(3/2)   +  j  2(1 + 4t^2)^(1/2)  -  (8t^2)(1 + 4t^2)^(-1/2))/(1 + 4t^2)

Multiply the second term (j) by (1 + 4t^2)^(1/2)/(1 + 4t^2)^(1/2), This makes both denominators, (i) and (j) equal and one term becomes simpler (-8t^2).

 

= ( -4t i  +   j ( 2(1 + 4t^2)  -  8t^2))/(1 + 4t^2)^(3/2)

= ( -4t i  +  2 j )/(1 + 4t^2)^(3/2)

 

N(t) = (-4t i  +  2 j)/(1  +  4t^2)^(1/2)