Question

Calculus help needed!

Without evaluating derivatives, which of the functions f(x)=ln x, g(x)=ln 11x, h(x)=lnx^2, and p(x)=ln 3x^2 have the same derivatives?
Multiple choice:
A. f'(x)=g'(x)
B. f'(x)=h'(x)
C. f'(x)=p'(x)
D. g'(x)=p'(x)
E. h'(x)=p'(x)
F. g'(x)=h'(x)
G. None of the derivatives are the same.
THANK YOU!

Answer 1

`f(x)=\ln x\\\\g(x)=\ln11x=\ln11+\ln x\ \ (\ln11=const.)\\\\h(x)=\ln x^2=2\ln x\\\\p(x)=\ln3x^2=\ln 3+\ln x^2=\ln3+2\ln x\ \ (\ln3=const.)`

Used:

`\ln a+\ln b=\ln(ab)\\\\\ln a^n=n\log a`

Functions have the same derivatives if they differ a constant.

h(x) = f(x); g(x) = f(x) + const.

h'(x) = f'(x)

g'(x) = (f(x) + const.)' = f'(x) + (const.)' = f'(x) + 0 = f'(x)

h'(x) = g'(x)

Therefore yuor answer is A. f'(x)=g'(x)

`f'(x)=(\ln x)'=(1)/(x)\\\\g'(x)=(\ln11x)=(1)/(11x)\cdot11=(1)/(x)`

Used:

`(\ln x)'=(1)/(x)\\\\f'(g(x))=f'(g(x))\cdot g'(x)`