Differentiate each of the following with respect to x in following question: `(3x+4)/(5x^(2)-7x+9)`

Answer

To differentiate the function \( y = \frac{3x + 4}{5x^2 - 7x + 9} \) with respect to \( x \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative is given by:

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

where \( u = 3x + 4 \) and \( v = 5x^2 - 7x + 9 \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = 3x + 4 \)
- \( v = 5x^2 - 7x + 9 \)

### Step 2: Differentiate \( u \) and \( v \)
Now we differentiate \( u \) and \( v \):
- \( \frac{du}{dx} = 3 \) (the derivative of \( 3x \) is \( 3 \) and the derivative of \( 4 \) is \( 0 \))
- \( \frac{dv}{dx} = 10x - 7 \) (the derivative of \( 5x^2 \) is \( 10x \), the derivative of \( -7x \) is \( -7 \), and the derivative of \( 9 \) is \( 0 \))

### Step 3: Apply the Quotient Rule
Now we apply the quotient rule:
\[ \frac{dy}{dx} = \frac{(5x^2 - 7x + 9)(3) - (3x + 4)(10x - 7)}{(5x^2 - 7x + 9)^2} \]

### Step 4: Simplify the numerator
Now we will simplify the numerator:
1. First, calculate \( (5x^2 - 7x + 9)(3) = 15x^2 - 21x + 27 \)
2. Next, calculate \( (3x + 4)(10x - 7) \):
- \( 3x \cdot 10x = 30x^2 \)
- \( 3x \cdot (-7) = -21x \)
- \( 4 \cdot 10x = 40x \)
- \( 4 \cdot (-7) = -28 \)
- Therefore, \( (3x + 4)(10x - 7) = 30x^2 + 19x - 28 \)

Now substitute back into the numerator:
\[ \frac{dy}{dx} = \frac{(15x^2 - 21x + 27) - (30x^2 + 19x - 28)}{(5x^2 - 7x + 9)^2} \]

### Step 5: Combine like terms
Now combine the like terms in the numerator:
\[ 15x^2 - 30x^2 - 21x - 19x + 27 + 28 = -15x^2 - 40x + 55 \]

### Step 6: Final expression
Thus, we have:
\[ \frac{dy}{dx} = \frac{-15x^2 - 40x + 55}{(5x^2 - 7x + 9)^2} \]

### Final Answer
The derivative of \( y = \frac{3x + 4}{5x^2 - 7x + 9} \) is:
\[ \frac{dy}{dx} = \frac{-15x^2 - 40x + 55}{(5x^2 - 7x + 9)^2} \]