To find the derivative of the given function $y = \left( \frac{3x}{7x + 6} \right)^5$, we'll use the chain rule and the quotient rule.

Step 1: Rewrite the function

Let: $u = \frac{3x}{7x + 6}$ Then the function becomes: $y = u^5$

Step 2: Differentiate $y$ with respect to $u$ using the chain rule

$\frac{dy}{du} = 5u^4$

Step 3: Differentiate $u = \frac{3x}{7x + 6}$ with respect to $x$ using the quotient rule

The quotient rule states: $\frac{d}{dx} \left( \frac{f}{g} \right) = \frac{f' \cdot g - f \cdot g'}{g^2}$ Let:

• $f = 3x$ and $f' = 3$
• $g = 7x + 6$ and $g' = 7$

Applying the quotient rule: $\frac{du}{dx} = \frac{(3)(7x + 6) - (3x)(7)}{(7x + 6)^2}$ Simplify the numerator: $(3)(7x + 6) - (3x)(7) = 21x + 18 - 21x = 18$ So: $\frac{du}{dx} = \frac{18}{(7x + 6)^2}$

Step 4: Combine using the chain rule

Using $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$: $\frac{dy}{dx} = 5u^4 \cdot \frac{18}{(7x + 6)^2}$ Substitute $u = \frac{3x}{7x + 6}$ back into the equation: $\frac{dy}{dx} = 5 \left( \frac{3x}{7x + 6} \right)^4 \cdot \frac{18}{(7x + 6)^2}$ $\frac{dy}{dx} = \frac{90x^4}{(7x + 6)^6}$

Final Answer

$\frac{dy}{dx} = \frac{90x^4}{(7x + 6)^6}$

Would you like more details or have any questions about this solution?

Here are five related questions to expand your understanding:

1. How does the chain rule work when differentiating composite functions?
2. What is the quotient rule and how is it applied to different functions?
3. How can we simplify expressions when applying the quotient rule?
4. How does the power rule interact with the chain rule in differentiating powers of a function?
5. What are some common mistakes to avoid when using the chain rule and quotient rule together?

Tip: When dealing with composite functions, it's always a good idea to first identify the inner and outer functions to apply the chain rule effectively.