If `y=log_(x^(2)+4)(7x^(2)-5x+1)`, then `(dy)/(dx)` is equal to

Options

A. `log_(e)(x^(2)+4).{(14x-5)/(7x^(2)-5x+1)-(2xy)/(x^(2)+4)}`
B. `(1)/(log_(e)(x^(2)+4)){(14x-5)/(7x^(2)-5x+1)-(2xy)/(x^(2)+4)}`
C. `log_(e)(7x^(2)-5x+1){(2x)/(x^(2)+4)-((14x-5)y)/(7x^(2)-5x+1)}`
D. `(1)/(log_(e)(7x^(2)-5x+1)){(2x)/(x^(2)+4)-((14x-5)y)/(7x^(2)-5x+1)}`

Answer

To find the derivative of the function \( y = \log_{(x^2 + 4)}(7x^2 - 5x + 1) \), we will use the change of base formula and the quotient rule for differentiation. Let's go through the steps:

### Step 1: Rewrite the logarithm using the change of base formula
Using the change of base formula, we can express the logarithm as:
\[ y = \frac{\log(7x^2 - 5x + 1)}{\log(x^2 + 4)} \]

### Step 2: Differentiate using the quotient rule
Let \( f(x) = \log(7x^2 - 5x + 1) \) and \( g(x) = \log(x^2 + 4) \). We will apply the quotient rule:
\[ \frac{dy}{dx} = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \]

### Step 3: Find \( f'(x) \) and \( g'(x) \)
1. Differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}[\log(7x^2 - 5x + 1)] = \frac{1}{7x^2 - 5x + 1} \cdot (14x - 5) \]
Thus,
\[ f'(x) = \frac{14x - 5}{7x^2 - 5x + 1} \]

2. Differentiate \( g(x) \):
\[ g'(x) = \frac{d}{dx}[\log(x^2 + 4)] = \frac{1}{x^2 + 4} \cdot (2x) \]
Thus,
\[ g'(x) = \frac{2x}{x^2 + 4} \]

### Step 4: Substitute \( f(x) \), \( g(x) \), \( f'(x) \), and \( g'(x) \) into the quotient rule
Now substituting back into the quotient rule:
\[ \frac{dy}{dx} = \frac{\left(\frac{14x - 5}{7x^2 - 5x + 1}\right) \log(x^2 + 4) - \log(7x^2 - 5x + 1) \left(\frac{2x}{x^2 + 4}\right)}{(\log(x^2 + 4))^2} \]

### Step 5: Simplify the expression
This expression can be simplified further, but the key components are already present. The final derivative is:
\[ \frac{dy}{dx} = \frac{(14x - 5) \log(x^2 + 4) - 2x \log(7x^2 - 5x + 1)}{(7x^2 - 5x + 1)(x^2 + 4)(\log(x^2 + 4))^2} \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = \frac{(14x - 5) \log(x^2 + 4) - 2x \log(7x^2 - 5x + 1)}{(7x^2 - 5x + 1)(x^2 + 4)(\log(x^2 + 4))^2} \]