Determine a process for finding g′ if g(x)= f(2x)/ f(x).

Answer 1

Use the quotient rule, (u/v)' = (u'v - uv')/v^2, and the chain rule (u(v(x)))' = u'(v(x))•v'(x)

So for g(x) = f(2x)/f(x) then u = f(2x) meaning u' = f '(2x)•2 = 2f '(2x) and v = f(x) meaning v' = f '(x) and v² = (f(x))²

Just put it all together and simplify as desired.

Answer 2

g(x) = f(2x)/f(x)

quotient rule: derivative of a quotient = the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared

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

then replace f(2x)/f(x) by g(x)

= [f'(2x) - g(x)f'(x)]/f(x)