If `5f(x)+3f(1/x)=x+2` and `y=x f(x),` then find `dy/dx` at `x=1`.

Options

A. 14
B. `(7)/(8)`
C. 1
D. none of these

Explanation

To solve the problem step by step, we will follow the instructions given in the video transcript while elaborating on each step.

### Step 1: Analyze the given equation
We start with the equation:
\[ 5f(x) + 3f\left(\frac{1}{x}\right) = x + 2 \]
This equation holds for all \( x \neq 0 \).

### Step 2: Substitute \( x \) with \( \frac{1}{x} \)
Next, we replace \( x \) with \( \frac{1}{x} \):
\[ 5f\left(\frac{1}{x}\right) + 3f(x) = \frac{1}{x} + 2 \]

### Step 3: Set up a system of equations
Now we have two equations:
1. \( 5f(x) + 3f\left(\frac{1}{x}\right) = x + 2 \) (Equation 1)
2. \( 5f\left(\frac{1}{x}\right) + 3f(x) = \frac{1}{x} + 2 \) (Equation 2)

### Step 4: Solve the system of equations
We can eliminate \( f\left(\frac{1}{x}\right) \) and \( f(x) \) by multiplying Equation 1 by 5 and Equation 2 by 3:
- From Equation 1:
\[ 25f(x) + 15f\left(\frac{1}{x}\right) = 5x + 10 \]
- From Equation 2:
\[ 15f\left(\frac{1}{x}\right) + 9f(x) = 3 + 6 \]

Now we can subtract these equations to eliminate \( f\left(\frac{1}{x}\right) \):
\[ (25f(x) + 15f\left(\frac{1}{x}\right)) - (15f\left(\frac{1}{x}\right) + 9f(x)) = (5x + 10) - (3 + 6) \]
This simplifies to:
\[ 16f(x) = 5x + 1 \]
Thus,
\[ f(x) = \frac{5x + 1}{16} \]

### Step 5: Find \( f(1) \)
Now we need to find \( f(1) \):
\[ f(1) = \frac{5(1) + 1}{16} = \frac{6}{16} = \frac{3}{8} \]

### Step 6: Differentiate \( f(x) \)
Next, we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}\left(\frac{5x + 1}{16}\right) = \frac{5}{16} \]

### Step 7: Define \( y \) and find \( \frac{dy}{dx} \)
Given \( y = x f(x) \):
\[ y = x \cdot \frac{5x + 1}{16} \]
Now, we differentiate \( y \):
\[ \frac{dy}{dx} = f(x) + x f'(x) \]
Substituting \( f(1) \) and \( f'(1) \):
\[ \frac{dy}{dx} = f(1) + 1 \cdot f'(1) = \frac{3}{8} + \frac{5}{16} \]

### Step 8: Find a common denominator and simplify
To add these fractions, we need a common denominator:
\[ \frac{3}{8} = \frac{6}{16} \]
Thus,
\[ \frac{dy}{dx} = \frac{6}{16} + \frac{5}{16} = \frac{11}{16} \]

### Final Answer
The value of \( \frac{dy}{dx} \) at \( x = 1 \) is:
\[ \frac{dy}{dx} = \frac{11}{16} \]