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: Understanding the Concept:
This problem involves a functional equation. We are asked to find a relationship between the derivatives based on the given equation. The key is to differentiate the functional equation with respect to \( x \) and then manipulate the result to find the desired ratio.
Step 2: Key Formula or Approach:
We will differentiate the equation \( (f(x))^n = f(nx) \) implicitly with respect to \( x \), using the chain rule on both sides.
- Left side: \( \frac{d}{dx} (f(x))^n = n(f(x))^{n-1} \cdot f'(x) \).
- Right side: \( \frac{d}{dx} f(nx) = f'(nx) \cdot \frac{d}{dx}(nx) = f'(nx) \cdot n \).
Step 3: Detailed Explanation:
Given the functional equation:
\[ (f(x))^n = f(nx) \] Differentiate both sides with respect to \( x \):
\[ \frac{d}{dx} \left( (f(x))^n \right) = \frac{d}{dx} \left( f(nx) \right) \] Applying the chain rule to both sides as described above:
\[ n \cdot (f(x))^{n-1} \cdot f'(x) = f'(nx) \cdot n \] Assuming \( n \neq 0 \), we can divide both sides by \( n \):
\[ (f(x))^{n-1} \cdot f'(x) = f'(nx) \] The question asks for the ratio \( \frac{f'(nx)}{f'(x)} \). We can rearrange the equation above to find this ratio (assuming \( f'(x) \neq 0 \)):
\[ \frac{f'(nx)}{f'(x)} = (f(x))^{n-1} \] This result is not directly among the options. We need to express \( (f(x))^{n-1} \) in terms of the given options using the original functional equation.
From the original equation \( (f(x))^n = f(nx) \), we can write:
\[ \frac{f(nx)}{f(x)} = \frac{(f(x))^n}{f(x)} = (f(x))^{n-1} \] Comparing the two results we derived:
\[ \frac{f'(nx)}{f'(x)} = (f(x))^{n-1} \quad \text{and} \quad \frac{f(nx)}{f(x)} = (f(x))^{n-1} \] Therefore, we can conclude:
\[ \frac{f'(nx)}{f'(x)} = \frac{f(nx)}{f(x)} \] Step 4: Final Answer:
The ratio \( \frac{f'(nx)}{f'(x)} \) is equal to \( \frac{f(nx)}{f(x)} \). This corresponds to option (C).