Given f(x)=4⋅arcsin(e^(8x)), find f′(x).

To find the derivative \( f'(x) \) of the function \( f(x) = 4 \cdot \arcsin(e^{8x}) \), we will apply the chain rule.

1. **Differentiate the outer function**: The derivative of \( \arcsin(u) \) is \( \frac{1}{\sqrt{1 - u^2}} \) where \( u = e^{8x} \).

2. **Differentiate the inner function**: The derivative of \( e^{8x} \) is \( 8e^{8x} \).

Now, using the chain rule:

\[
f'(x) = 4 \cdot \frac{1}{\sqrt{1 - (e^{8x})^2}} \cdot (8e^{8x})
\]

Thus, we can write:

\[
f'(x) = \frac{32 e^{8x}}{\sqrt{1 - e^{16x}}}
\]

This is the derivative of the function.

### Summary
\[
f'(x) = \frac{32 e^{8x}}{\sqrt{1 - e^{16x}}}
\]

Would you like to see a detailed explanation of any steps? Here are some follow-up questions:

1. What is the significance of the chain rule in calculus?
2. Can you explain the properties of the arcsine function?
3. How do you evaluate the derivative at a specific point?
4. What happens to \( f'(x) \) as \( x \) approaches certain limits?
5. Can you give examples of other functions involving arcsine? 

**Tip**: Always check if the inner function is within the domain of the arcsine to avoid undefined values.

Given f(x)=4⋅arcsin(e^(8x)), find f′(x).

Explore how to find the derivative f'(x) for the function f(x)=4⋅arcsin(e^(8x)). This solution utilizes the chain rule and includes key mathematical concepts relevant to calculus, making it suitable for high school students in Grades 11-12.

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