Let `y= x+(1)/(x)`.Find the rate of change of y w.r.t. x at x=2.

Answer

To find the rate of change of \( y \) with respect to \( x \) at \( x = 2 \) for the function \( y = x + \frac{1}{x} \), we will follow these steps:

### Step 1: Write down the function
We are given:
\[ y = x + \frac{1}{x} \]

### Step 2: Differentiate \( y \) with respect to \( x \)
To find the rate of change, we need to differentiate \( y \):
\[ \frac{dy}{dx} = \frac{d}{dx} \left( x + \frac{1}{x} \right) \]

### Step 3: Apply the differentiation rules
Using the sum rule of differentiation, we differentiate each term:
- The derivative of \( x \) is \( 1 \).
- The derivative of \( \frac{1}{x} \) can be rewritten as \( x^{-1} \). Using the power rule, we have:
\[ \frac{d}{dx} \left( x^{-1} \right) = -1 \cdot x^{-2} = -\frac{1}{x^2} \]
Thus, we have:
\[ \frac{dy}{dx} = 1 - \frac{1}{x^2} \]

### Step 4: Evaluate the derivative at \( x = 2 \)
Now we substitute \( x = 2 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x=2} = 1 - \frac{1}{2^2} = 1 - \frac{1}{4} \]
Calculating this gives:
\[ \frac{dy}{dx} \bigg|_{x=2} = 1 - 0.25 = 0.75 = \frac{3}{4} \]

### Final Answer
The rate of change of \( y \) with respect to \( x \) at \( x = 2 \) is:
\[ \frac{3}{4} \]

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