Find the derivative of :
`f(x)=(2x+3)/(x-2)`

Answer

To find the derivative of the function \( f(x) = \frac{2x + 3}{x - 2} \), we will use the quotient rule for differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative \( f'(x) \) is given by:

\[ f'(x) = \frac{u'v - uv'}{v^2} \]

where \( u \) is the numerator and \( v \) is the denominator.

### Step 1: Identify \( u \) and \( v \)

Let:
- \( u = 2x + 3 \)
- \( v = x - 2 \)

### Step 2: Find the derivatives \( u' \) and \( v' \)

Now, we need to find the derivatives of \( u \) and \( v \):
- \( u' = \frac{d}{dx}(2x + 3) = 2 \)
- \( v' = \frac{d}{dx}(x - 2) = 1 \)

### Step 3: Apply the quotient rule

Now we can apply the quotient rule:

\[ f'(x) = \frac{u'v - uv'}{v^2} \]

Substituting the values we found:

\[ f'(x) = \frac{(2)(x - 2) - (2x + 3)(1)}{(x - 2)^2} \]

### Step 4: Simplify the numerator

Now, let's simplify the numerator:

\[ f'(x) = \frac{2(x - 2) - (2x + 3)}{(x - 2)^2} \]

Expanding the terms in the numerator:

\[ = \frac{2x - 4 - 2x - 3}{(x - 2)^2} \]

Combining like terms:

\[ = \frac{-4 - 3}{(x - 2)^2} = \frac{-7}{(x - 2)^2} \]

### Final Result

Thus, the derivative of the function \( f(x) \) is:

\[ f'(x) = \frac{-7}{(x - 2)^2} \]

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