Question

If $(f(x))^n=f(nx)$, then $\frac{f'(nx)}{f'(x)}$ is:

• A. \((f(x))^n\)
• B. \(n f(nx)\)
• C. \(\frac{f(nx)}{f(x)}\)
• D. \(\frac{f((n-1)x)}{f(x)}\)
• E. \(\frac{f(x)}{f(nx)}\)

Hint:

When a function is given in power form, differentiate both sides and then use the original relation again to simplify the result.

Answer 1

The Correct Option is C

Step 1: Write the given functional relation.
\[ (f(x))^n=f(nx) \]

Step 2: Differentiate both sides.
Differentiate with respect to \(x\):
\[ n(f(x))^{n-1}f'(x)=f'(nx)\cdot n \] (using chain rule on RHS)

Step 3: Simplify the equation.
Cancel \(n\) from both sides: \[ (f(x))^{n-1}f'(x)=f'(nx) \]

Step 4: Rearrange for required ratio.
\[ \frac{f'(nx)}{f'(x)}=(f(x))^{n-1} \]

Step 5: Express in terms of given relation.
From original equation: \[ (f(x))^n=f(nx) \] So, \[ (f(x))^{n-1}=\frac{f(nx)}{f(x)} \]

Step 6: Substitute into result.
\[ \frac{f'(nx)}{f'(x)}=\frac{f(nx)}{f(x)} \]

Step 7: Final answer.
\[ \boxed{\frac{f(nx)}{f(x)}} \] which matches option \((3)\).