We are asked to find the derivative of the following function:

$h(x) = \frac{(4x - 4)^5}{(2x - 5)^7}$

To differentiate this, we will apply the quotient rule. The quotient rule states that if we have a function of the form $\frac{f(x)}{g(x)}$, the derivative is:

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

Step-by-Step Solution:

Let:

• $f(x) = (4x - 4)^5$
• $g(x) = (2x - 5)^7$

1. Compute $f'(x)$:

We apply the chain rule to differentiate $f(x)$: $f'(x) = 5(4x - 4)^4 \cdot \frac{d}{dx}(4x - 4)$ $f'(x) = 5(4x - 4)^4 \cdot 4 = 20(4x - 4)^4$

2. Compute $g'(x)$:

Again, apply the chain rule to $g(x)$: $g'(x) = 7(2x - 5)^6 \cdot \frac{d}{dx}(2x - 5)$ $g'(x) = 7(2x - 5)^6 \cdot 2 = 14(2x - 5)^6$

3. Apply the quotient rule:

Now substitute into the quotient rule formula: $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$ Substitute $f(x), g(x), f'(x), g'(x)$: $h'(x) = \frac{20(4x - 4)^4 (2x - 5)^7 - (4x - 4)^5 14(2x - 5)^6}{(2x - 5)^{14}}$

4. Simplify:

Factor out common terms from the numerator: $h'(x) = \frac{(4x - 4)^4 (2x - 5)^6 \left[ 20(2x - 5) - 14(4x - 4) \right]}{(2x - 5)^{14}}$ Now simplify the expression inside the brackets: $20(2x - 5) - 14(4x - 4) = 40x - 100 - 56x + 56 = -16x - 44 = -4(4x + 11)$ Substitute this back: $h'(x) = \frac{(4x - 4)^4 (2x - 5)^6 (-4)(4x + 11)}{(2x - 5)^{14}}$

Finally, simplify the powers of $(2x - 5)$: $h'(x) = \frac{-(4x - 4)^4 (4x + 11)}{(2x - 5)^8}$

Final Answer:

$h'(x) = \frac{-(4x - 4)^4 (4x + 11)}{(2x - 5)^8}$

Let me know if you'd like further details or explanations!

Here are some related questions for practice:

1. How do you apply the chain rule in complex functions?
2. What is the product rule, and how does it compare to the quotient rule?
3. How can we further simplify expressions involving derivatives?
4. Why is it important to leave the derivative in factored form?
5. How does the power rule apply in differentiation?

Tip: Always look for common factors when simplifying derivatives—it often helps reduce the complexity of the expression.