`y= (2x+5)/(3x-2)`

Options

A. `y'=(-19)/((3x-2))^(2)`
B. `y'=(19)/(3x-2)^(2)`
C. `y'=(19)/(3x+2)^(2)`
D. `y'=(-19)/(3x+2)^(2)`

Answer

To find the derivative of the function \( y = \frac{2x + 5}{3x - 2} \), we will use the quotient rule. The quotient rule states that if you have a function in the form of \( \frac{u}{v} \), then the derivative \( \frac{dy}{dx} \) is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where:
- \( u = 2x + 5 \)
- \( v = 3x - 2 \)

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

Let:
- \( u = 2x + 5 \)
- \( v = 3x - 2 \)

### Step 2: Differentiate \( u \) and \( v \)

Now we need to find the derivatives of \( u \) and \( v \):

- \( \frac{du}{dx} = \frac{d}{dx}(2x + 5) = 2 \)
- \( \frac{dv}{dx} = \frac{d}{dx}(3x - 2) = 3 \)

### Step 3: Apply the Quotient Rule

Now, substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:

\[ \frac{dy}{dx} = \frac{(3x - 2)(2) - (2x + 5)(3)}{(3x - 2)^2} \]

### Step 4: Simplify the Numerator

Now simplify the numerator:

1. Calculate \( (3x - 2)(2) = 6x - 4 \)
2. Calculate \( (2x + 5)(3) = 6x + 15 \)

So, the numerator becomes:

\[ 6x - 4 - (6x + 15) = 6x - 4 - 6x - 15 = -4 - 15 = -19 \]

### Step 5: Write the Final Derivative

Now we can write the final expression for the derivative:

\[ \frac{dy}{dx} = \frac{-19}{(3x - 2)^2} \]

### Final Answer

Thus, the derivative of \( y = \frac{2x + 5}{3x - 2} \) is:

\[ \frac{dy}{dx} = \frac{-19}{(3x - 2)^2} \]