Let f be a differential function satisfying the condition. `f((x)/(y))=(f(x))/(f(y))"for all "x,y ( ne 0) in R"and f(y) ne 0` If f'(1)=2`, then f'(x) is equal to

Options

A. 2f(x)
B. `(f(x))/(2)`
C. 2x f(x)
D. `(2f(x))/(x)`

Explanation

To solve the problem, we need to analyze the given functional equation and use the information provided about the derivative at a specific point. Here’s the step-by-step solution:

### Step 1: Analyze the Functional Equation
We are given the functional equation:
\[ f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)} \]
for all \(x, y \neq 0\) and \(f(y) \neq 0\).

### Step 2: Substitute Specific Values
Let’s substitute \(x = 1\) and \(y = 1\):
\[ f\left(\frac{1}{1}\right) = \frac{f(1)}{f(1)} \]
This simplifies to:
\[ f(1) = 1 \]

### Step 3: Differentiate the Functional Equation
To find \(f'(x)\), we can differentiate both sides of the functional equation with respect to \(x\). Using the quotient rule and chain rule, we differentiate:
\[ \frac{d}{dx}\left[f\left(\frac{x}{y}\right)\right] = \frac{d}{dx}\left[\frac{f(x)}{f(y)}\right] \]
The right-hand side becomes:
\[ \frac{f'(x)f(y) - f(x)f'(y) \cdot 0}{f(y)^2} = \frac{f'(x)}{f(y)} \]

### Step 4: Evaluate at \(y = 1\)
Now, let’s evaluate this at \(y = 1\):
\[ f'\left(\frac{x}{1}\right) = \frac{f'(x)}{f(1)} = f'(x) \]
Thus, we have:
\[ f'(x) = f'\left(x\right) \]

### Step 5: Use Given Information
We know that \(f'(1) = 2\). From our earlier result, we can establish:
\[ f'(x) = k \cdot f(x) \]
for some constant \(k\). Since \(f'(1) = 2\) and \(f(1) = 1\), we have:
\[ 2 = k \cdot 1 \implies k = 2 \]

### Step 6: General Form of \(f'(x)\)
Thus, we can conclude:
\[ f'(x) = 2f(x) \]

### Step 7: Conclusion
The final result is:
\[ f'(x) = 2f(x) \]