To find the derivative of the given function

$y = \left( \frac{x - 4}{2x + 3} \right)^7,$

we will use the chain rule combined with the quotient rule for differentiation.

Step 1: General Chain Rule

The derivative of $y = u^7$, where $u = \frac{x - 4}{2x + 3}$, is:

$\frac{dy}{dx} = 7u^6 \cdot \frac{du}{dx}.$

So, we need to find $\frac{du}{dx}$, where $u = \frac{x - 4}{2x + 3}$.

Step 2: Quotient Rule for $u = \frac{x - 4}{2x + 3}$

The quotient rule states:

$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}.$

Here:

• $f(x) = x - 4$, so $f'(x) = 1$,
• $g(x) = 2x + 3$, so $g'(x) = 2$.

Substitute these into the quotient rule:

$\frac{du}{dx} = \frac{(1)(2x + 3) - (x - 4)(2)}{(2x + 3)^2}.$

Simplify the numerator:

$(1)(2x + 3) - (x - 4)(2) = 2x + 3 - 2x + 8 = 11.$

Thus:

$\frac{du}{dx} = \frac{11}{(2x + 3)^2}.$

Step 3: Combine Results

Substitute $u = \frac{x - 4}{2x + 3}$ and $\frac{du}{dx}$ into the chain rule:

$\frac{dy}{dx} = 7 \left( \frac{x - 4}{2x + 3} \right)^6 \cdot \frac{11}{(2x + 3)^2}.$

Step 4: Simplify the Expression

Combine terms:

$\frac{dy}{dx} = \frac{7 \cdot 11 \cdot (x - 4)^6}{(2x + 3)^{6 + 2}} = \frac{77 (x - 4)^6}{(2x + 3)^8}.$

Final Answer:

$\frac{dy}{dx} = \frac{77 (x - 4)^6}{(2x + 3)^8}.$

Would you like further steps or additional examples to clarify?

Here are some follow-up questions for practice:

1. How does the chain rule apply to composite functions?
2. How do you differentiate a function with a fractional exponent?
3. Can you find the second derivative of the given function?
4. What is the behavior of the derivative as $x \to \infty$?
5. Can you graph the original function and its derivative?

Tip: When differentiating functions with fractions, simplify each step to avoid errors in algebra.