If f(x)=4sec(8x)f(x)=4sec(8x), find f'(x)f′(x).

Find f'(4)f′(4).

### Step 1: Find \( f'(x) \)
Given the function \( f(x) = 4 \sec(8x) \), we need to find its derivative, \( f'(x) \).

1. The derivative of \( \sec(u) \) is \( \sec(u) \tan(u) \cdot \frac{du}{dx} \), where \( u \) is a function of \( x \).
2. In our case, \( u = 8x \), so we apply the chain rule.

First, differentiate the outer function:
\[
f'(x) = 4 \cdot \frac{d}{dx}[\sec(8x)]
\]
\[
= 4 \cdot \sec(8x) \tan(8x) \cdot \frac{d}{dx}[8x]
\]
Now differentiate \( 8x \):
\[
\frac{d}{dx}[8x] = 8
\]
So,
\[
f'(x) = 4 \cdot \sec(8x) \tan(8x) \cdot 8
\]
\[
f'(x) = 32 \sec(8x) \tan(8x)
\]

### Step 2: Find \( f'(4) \)
Now, substitute \( x = 4 \) into the expression for \( f'(x) \):
\[
f'(4) = 32 \sec(8(4)) \tan(8(4))
\]
\[
f'(4) = 32 \sec(32) \tan(32)
\]

Therefore, \( f'(4) = 32 \sec(32^\circ) \tan(32^\circ) \) (assuming degrees, unless radians are specified).

Do you want more details on how to evaluate this expression, or have any other questions?

Here are some related questions:
1. How do you differentiate \( \sin(8x) \) or \( \cos(8x) \)?
2. What are the general rules for differentiating trigonometric functions?
3. Can you explain the chain rule more deeply?
4. How do we evaluate trigonometric functions at specific angles?
5. What is the relationship between secant, cosine, and their derivatives?

**Tip:** When differentiating composite functions, always remember to apply the chain rule.

If f(x) = 4 sec(8x), find f'(x). Find f'(4).

This problem involves finding the derivative of the function f(x) = 4 sec(8x) using the chain rule and basic trigonometric derivatives. The step-by-step solution shows how to apply the chain rule and evaluate the result at x = 4. Suitable for students in Grades 11-12 or early calculus courses.

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