The given function is:

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

To find the derivative $h'(x)$, we will apply the quotient rule, which states:

$\left( \frac{u(x)}{v(x)} \right)' = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$

Here,

• $u(x) = (4x - 4)^5$
• $v(x) = (2x - 5)^7$

Step 1: Find $u'(x)$ and $v'(x)$

For $u(x) = (4x - 4)^5$, we use the chain rule: $u'(x) = 5(4x - 4)^4 \cdot \frac{d}{dx}(4x - 4) = 5(4x - 4)^4 \cdot 4 = 20(4x - 4)^4$

For $v(x) = (2x - 5)^7$, again using the chain rule: $v'(x) = 7(2x - 5)^6 \cdot \frac{d}{dx}(2x - 5) = 7(2x - 5)^6 \cdot 2 = 14(2x - 5)^6$

Step 2: Apply the quotient rule

Now we apply the quotient rule:

$h'(x) = \frac{20(4x - 4)^4 (2x - 5)^7 - (4x - 4)^5 \cdot 14(2x - 5)^6}{(2x - 5)^{14}}$

Step 3: Factor the numerator

To simplify the numerator, factor out the common terms:

$h'(x) = \frac{(4x - 4)^4 (2x - 5)^6 \left[ 20(2x - 5) - 14(4x - 4) \right]}{(2x - 5)^{14}}$

Simplify the expression inside the brackets: $20(2x - 5) - 14(4x - 4) = 40x - 100 - 56x + 56 = -16x - 44$ Thus, the numerator becomes:

$h'(x) = \frac{(4x - 4)^4 (2x - 5)^6 (-16x - 44)}{(2x - 5)^{14}}$

Step 4: Simplify the final expression

Now simplify the denominator: $h'(x) = \frac{(4x - 4)^4 (-16x - 44)}{(2x - 5)^8}$

This is the derivative $h'(x)$ in its factored form, simplified as much as possible.

Do you have any questions about the steps, or would you like further details?

Here are some related questions you might find interesting:

1. How do you apply the quotient rule in other similar problems?
2. What is the significance of the chain rule when differentiating composite functions?
3. How can you simplify expressions involving powers after differentiation?
4. Can the derivative change when factoring terms in different ways?
5. What alternative methods can be used to find derivatives of rational functions?

Tip: When simplifying derivatives, always look for common factors to simplify the expressions efficiently.