The derivative of `f(x) = (3 x + 2 "sin" x)/(x + 5 cos x)` is

Options

A. `(cos x (2x + 15) + 10 + "sin" x (15x - 2))/((x + 5 cos x)^(2))`
B. `((cos x + 10)("sin" x - 15))/((x + 5 cos x)^(2))`
C. `(cos x (2x + 16) + 15 + "sin" x)/((x 5 cos x)^(2))`
D. `((2x + 15) (15 + "sin" x))/((x + 5 cos x)^(2))`

Answer

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

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

where \( u = 3x + 2 \sin x \) and \( v = x + 5 \cos x \).

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

1. **Differentiate \( u \)**:
\[ u = 3x + 2 \sin x \]
\[ u' = 3 + 2 \cos x \]

2. **Differentiate \( v \)**:
\[ v = x + 5 \cos x \]
\[ v' = 1 - 5 \sin x \]

### Step 2: Apply the Quotient Rule

Now, we can apply the quotient rule:

\[ f'(x) = \frac{(3 + 2 \cos x)(x + 5 \cos x) - (3x + 2 \sin x)(1 - 5 \sin x)}{(x + 5 \cos x)^2} \]

### Step 3: Simplify the Numerator

Now, we will simplify the numerator:

1. **Expand the first term**:
\[ (3 + 2 \cos x)(x + 5 \cos x) = 3x + 15 \cos x + 2x \cos x + 10 \cos^2 x \]

2. **Expand the second term**:
\[ (3x + 2 \sin x)(1 - 5 \sin x) = 3x - 15x \sin x + 2 \sin x - 10 \sin^2 x \]

3. **Combine the two expansions**:
\[ \text{Numerator} = (3x + 15 \cos x + 2x \cos x + 10 \cos^2 x) - (3x - 15x \sin x + 2 \sin x - 10 \sin^2 x) \]

This simplifies to:
\[ 3x + 15 \cos x + 2x \cos x + 10 \cos^2 x - 3x + 15x \sin x - 2 \sin x + 10 \sin^2 x \]

The \( 3x \) terms cancel out:
\[ = 15 \cos x + 2x \cos x + 10 \cos^2 x + 15x \sin x - 2 \sin x + 10 \sin^2 x \]

### Step 4: Final Expression

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

\[ f'(x) = \frac{15 \cos x + 2x \cos x + 10 \cos^2 x + 15x \sin x - 2 \sin x + 10 \sin^2 x}{(x + 5 \cos x)^2} \]