Find the derivative of the function.
y = ln 

4

x − 7
x + 7

 

To find the derivative of the function \( y = \ln \left( \frac{4x - 7}{x + 7} \right) \), we'll use the chain rule and the quotient rule.

### Step 1: Define the inner function
Let \( u = \frac{4x - 7}{x + 7} \).

### Step 2: Differentiate the natural logarithm function
The derivative of \( y = \ln(u) \) with respect to \( x \) is:
\[
\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}
\]

### Step 3: Differentiate the inner function \( u = \frac{4x - 7}{x + 7} \)
To differentiate \( u \), we'll apply the quotient rule, which states:
\[
\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
\]
Here, \( f(x) = 4x - 7 \) and \( g(x) = x + 7 \).

Compute the derivatives:
\[
f'(x) = 4
\]
\[
g'(x) = 1
\]

Apply the quotient rule:
\[
\frac{du}{dx} = \frac{(4)(x + 7) - (4x - 7)(1)}{(x + 7)^2}
\]

Simplify the numerator:
\[
\frac{du}{dx} = \frac{4x + 28 - 4x + 7}{(x + 7)^2} = \frac{35}{(x + 7)^2}
\]

### Step 4: Substitute back into the derivative of \( y \)
Now, substitute \( u = \frac{4x - 7}{x + 7} \) and \( \frac{du}{dx} = \frac{35}{(x + 7)^2} \) into the chain rule formula:
\[
\frac{dy}{dx} = \frac{1}{\frac{4x - 7}{x + 7}} \cdot \frac{35}{(x + 7)^2}
\]

### Step 5: Simplify the expression
\[
\frac{dy}{dx} = \frac{35(x + 7)}{(x + 7)^2 (4x - 7)} = \frac{35}{(x + 7)(4x - 7)}
\]

So, the derivative of the function is:
\[
\boxed{\frac{35}{(x + 7)(4x - 7)}}
\]

Would you like more details on any specific step or have any questions? Here are some related questions to explore further:

1. What is the chain rule in differentiation?
2. How do you apply the quotient rule to a given function?
3. How can we simplify complex fractions in calculus?
4. What is the derivative of the natural logarithm function?
5. How do you interpret the derivative of a logarithmic function graphically?

**Tip:** When applying the quotient rule, always ensure you correctly identify and differentiate both the numerator and denominator functions before substituting them into the rule.

Learn how to find the derivative of the natural logarithm function ln((4x - 7)/(x + 7)) using the chain rule and quotient rule. This problem explores calculus concepts such as derivatives, chain rule, and quotient rule. Suitable for college-level students and calculus enthusiasts.

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