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If \(f\) is a differentiable function such that \([f(x)]^n=f(nx)\) for every real number \(x\), what is the value of \(\frac{f'(x)}{f(x^n)}\)?

1. f(x)
2. 0
3. f(x)/(nx)
4. None of the above

Correct answer: f(x)/(nx)

Solution

Differentiating \([f(x)]^n=f(nx)\) gives \(n[f(x)]^{n-1}f'(x)=n f'(nx)\). Using the original relation to rewrite terms leads to the required expression. The option matching the derived form is \(\frac{f(x)}{nx}\).

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