Let `y= (x+3) /( x)`.Find the instantaneous rate of change of y with respect to x at x = 3.

Answer

To find the instantaneous rate of change of \( y \) with respect to \( x \) at \( x = 3 \) for the function \( y = \frac{x + 3}{x} \), we will follow these steps:

### Step 1: Simplify the function
We start with the function:
\[ y = \frac{x + 3}{x} \]
We can simplify this by separating the terms:
\[ y = \frac{x}{x} + \frac{3}{x} = 1 + \frac{3}{x} \]

### Step 2: Differentiate the function
Now, we need to find the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{d}{dx}(1 + \frac{3}{x}) \]
Since the derivative of a constant is 0, we focus on the second term:
\[ \frac{dy}{dx} = 0 + \frac{d}{dx}(3 \cdot x^{-1}) \]
Using the power rule for differentiation, where \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \):
\[ \frac{dy}{dx} = 3 \cdot (-1) \cdot x^{-2} = -\frac{3}{x^2} \]

### Step 3: Evaluate the derivative at \( x = 3 \)
Now we will substitute \( x = 3 \) into the derivative to find the instantaneous rate of change:
\[ \frac{dy}{dx} \bigg|_{x=3} = -\frac{3}{3^2} = -\frac{3}{9} = -\frac{1}{3} \]

### Final Answer
The instantaneous rate of change of \( y \) with respect to \( x \) at \( x = 3 \) is:
\[ -\frac{1}{3} \]

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