Find the derivative of the given function.

y equals ln StartFraction 7 x Over 3 plus x EndFractiony=ln7x3+x

To find the derivative of the function \( y = \ln\left(\frac{7x}{3 + x}\right) \), we'll apply the chain rule along with the quotient rule.

### Step 1: Simplify the function
First, let's rewrite the function:
\[
y = \ln\left(\frac{7x}{3 + x}\right)
\]

### Step 2: Differentiate using the chain rule
The derivative of \( y = \ln(u) \) with respect to \( x \) is \( \frac{1}{u} \cdot \frac{du}{dx} \).

Let \( u = \frac{7x}{3 + x} \). Then, the derivative of \( y \) is:
\[
\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}
\]
\[
\frac{dy}{dx} = \frac{1}{\frac{7x}{3 + x}} \cdot \frac{d}{dx}\left(\frac{7x}{3 + x}\right)
\]
\[
\frac{dy}{dx} = \frac{3 + x}{7x} \cdot \frac{d}{dx}\left(\frac{7x}{3 + x}\right)
\]

### Step 3: Differentiate using the quotient rule
Now, differentiate \( u = \frac{7x}{3 + x} \) using the quotient rule. 

The quotient rule states that if \( u = \frac{f(x)}{g(x)} \), then:
\[
\frac{du}{dx} = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
\]

Here, \( f(x) = 7x \) and \( g(x) = 3 + x \).
- \( f'(x) = 7 \)
- \( g'(x) = 1 \)

Using the quotient rule:
\[
\frac{du}{dx} = \frac{(7)(3 + x) - (7x)(1)}{(3 + x)^2}
\]
\[
\frac{du}{dx} = \frac{21 + 7x - 7x}{(3 + x)^2}
\]
\[
\frac{du}{dx} = \frac{21}{(3 + x)^2}
\]

### Step 4: Combine the results
Substitute \( \frac{du}{dx} \) back into the expression for \( \frac{dy}{dx} \):
\[
\frac{dy}{dx} = \frac{3 + x}{7x} \cdot \frac{21}{(3 + x)^2}
\]
Simplifying:
\[
\frac{dy}{dx} = \frac{21(3 + x)}{7x(3 + x)^2}
\]
\[
\frac{dy}{dx} = \frac{21}{7x(3 + x)}
\]
\[
\frac{dy}{dx} = \frac{3}{x(3 + x)}
\]

So, the derivative of the function is:
\[
\frac{dy}{dx} = \frac{3}{x(3 + x)}
\]

Would you like more details, or do you have any questions?

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Here are some related questions:

1. How would you differentiate \( y = \ln(x^2 + 1) \)?
2. What is the derivative of \( y = \ln\left(\frac{2x + 1}{x - 3}\right) \)?
3. How does the chain rule apply when differentiating nested logarithmic functions?
4. Can you find the derivative of \( y = \ln(5x + \sin(x)) \)?
5. How would you use the quotient rule to differentiate \( y = \frac{x^2 + 1}{x^2 - 1} \)?
6. What is the derivative of \( y = \ln\left(\frac{x^2 + 2x + 1}{x^2 + 4x + 4}\right) \)?
7. How would the derivative change if the function were \( y = \ln\left(\frac{7x}{3 + x^2}\right) \)?
8. How do logarithmic differentiation techniques apply to more complex functions?

**Tip:** When differentiating functions involving logarithms, always consider whether simplifying the function first will make the differentiation process easier.

Learn how to find the derivative of ln(7x/(3 + x)) using the chain rule and quotient rule. This problem involves logarithmic functions and advanced differentiation techniques. Suitable for advanced high school students and college-level calculus learners.

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