Question

use the fact that f(x)= U(x)/V(x) can be written as f(x)V(x) = U(x) and the product rule for derivatives to verify the quotient rule for derivatives.

Answer 1

f'(x)V(x)+V'(x)f(x)=U'(x)
f'(x)V(x)=U'(x)-V(x)f(x)
f'(x)V(x)=U'(x)-V'(x)(U(x)/V(x))
same denominator

f'(x)V(x)=[U'(x)V(x)-V'(x)U(x)]/V(x)

f'(x)=[U'(x)V(x)-V'(x)U(x)]/[V(x)V(x)]
f'(x)=[U'(x)V(x)-V'(x)U(x)]/[V(x)]^2

Answer 2

so basically you started with substitution and you just substituted U(x) in?

Answer 3

you have f(x)V(x)=U(x)
take derivative of both sides and solve for f'(x)
derivative of the left side requires product rule so you do that... :)