Find the derivative of `(9x^(5))/(x-3)` with respect to x.

Options

A. `(9x^(4)(4x-15))/((x+3)^(2))`
B. `(9x^(4)(4x-5))/((x-3)^(2))`
C. `(9x^(4)(4x+15))/((x-3)^(2))`
D. `(9x^(4)(4x+15))/((x+3)^(2))`

Answer

To find the derivative of the function \( f(x) = \frac{9x^5}{x-3} \) with respect to \( x \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( x \), then the derivative is given by:

\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

### Step 1: Identify \( u \) and \( v \)
In our case:
- \( u = 9x^5 \)
- \( v = x - 3 \)

### Step 2: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)
Now, we need to find the derivatives of \( u \) and \( v \):
- \( \frac{du}{dx} = \frac{d}{dx}(9x^5) = 9 \cdot 5x^{4} = 45x^4 \)
- \( \frac{dv}{dx} = \frac{d}{dx}(x - 3) = 1 \)

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:

\[ \frac{d}{dx} \left( \frac{9x^5}{x-3} \right) = \frac{(x - 3)(45x^4) - (9x^5)(1)}{(x - 3)^2} \]

### Step 4: Simplify the Expression
Now, we will simplify the numerator:

1. Expand the first term:
\[ (x - 3)(45x^4) = 45x^5 - 135x^4 \]

2. Combine it with the second term:
\[ 45x^5 - 135x^4 - 9x^5 = (45x^5 - 9x^5) - 135x^4 = 36x^5 - 135x^4 \]

3. Factor out \( 9x^4 \):
\[ = 9x^4(4x - 15) \]

So, the derivative becomes:
\[ \frac{9x^4(4x - 15)}{(x - 3)^2} \]

### Final Result
Thus, the derivative of \( f(x) = \frac{9x^5}{x-3} \) with respect to \( x \) is:
\[ f'(x) = \frac{9x^4(4x - 15)}{(x - 3)^2} \]

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