Let `f` be a differentiable function satisfying `[f(x)]^(n)=f(nx)` for all `x in R` Then `f'(x)f(nx)` equals

Options

A. `f(x)`
B. 0
C. `f(x)f'(nx)`
D. none of these

Answer

C

Explanation

We have,
`[f(x)]^(n)=f(nx)" for all "x`
`implies" "n[f(x)]^(n-1)f'(x)=nf'(nx)`
`implies" "n[f(x)]^(n)f'(x)=nf(x)f'(nx)" "[{:("Multiplying both sides"),(" by f(x)"):}]`
`implies" "nf(nx)f'(x)=nf(x)f,(nx)" "[because[f(x)]^(n)=f(nx)]`
`implies" "f(nx)f'(x)=f(x)f'(nx)`