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The problem provided requires us to differentiate the function \( h(x) = |7x| \cos(8x) \) using the given formula for the derivative of the absolute value.

### The formula given is:
\[
\frac{d}{dx} |u| = \frac{u'}{|u|}, \text{ for } u \neq 0.
\]

### Step-by-step solution:

1. **Let \( u(x) = 7x \).** 

The absolute value part \( |7x| \) involves differentiating using the provided formula:
\[
\frac{d}{dx} |7x| = \frac{7}{|7x|}.
\]

2. **Differentiate the second part \( \cos(8x) \):**

The derivative of \( \cos(8x) \) is:
\[
\frac{d}{dx} \cos(8x) = -8 \sin(8x).
\]

3. **Apply the product rule:**

Since \( h(x) = |7x| \cos(8x) \), we use the product rule for differentiation:
\[
h'(x) = \frac{d}{dx} \left( |7x| \right) \cdot \cos(8x) + |7x| \cdot \frac{d}{dx} \left( \cos(8x) \right).
\]
Substitute the derivatives:
\[
h'(x) = \frac{7}{|7x|} \cdot \cos(8x) + |7x| \cdot (-8 \sin(8x)).
\]

4. **Simplify the result:**
\[
h'(x) = \frac{7 \cos(8x)}{|x|} - 8 |7x| \sin(8x).
\]
Since \( |7x| = 7|x| \), we substitute that into the expression:
\[
h'(x) = \frac{7 \cos(8x)}{|x|} - 56 |x| \sin(8x).
\]

Thus, the derivative of \( h(x) = |7x| \cos(8x) \) is:
\[
h'(x) = \frac{7 \cos(8x)}{|x|} - 56 |x| \sin(8x).
\]

---

Would you like further details or clarifications on any of the steps? Here are some related questions:

1. What is the product rule and how is it applied?
2. How is the derivative of \( \cos(8x) \) computed?
3. Can you explain the absolute value derivative formula?
4. Why is \( |7x| \) treated as \( 7|x| \)?
5. How does the chain rule relate to this problem?

**Tip:** When differentiating functions involving absolute values, always pay attention to the cases when the function inside the absolute value changes signs!

Find the derivative of the function h(x) = |7x|cos(8x) using the given derivative formula for absolute value.

Learn how to find the derivative of h(x) = |7x|cos(8x) using the product rule, the chain rule, and the formula for the derivative of an absolute value. This detailed step-by-step solution explores key calculus concepts, including trigonometric derivatives and handling absolute values.

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