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The problem asks for the derivative of the function:

\[
f(x) = \sin^{-1}(14 \cos x)
\]

We need to differentiate this function with respect to \(x\).

### Step 1: Use the chain rule.

The function is composed of an outer function, the inverse sine \(\sin^{-1}(u)\), and an inner function \(u = 14 \cos(x)\). The chain rule tells us:

\[
f'(x) = \frac{d}{dx} \left[ \sin^{-1}(u) \right] \cdot \frac{du}{dx}
\]

The derivative of \(\sin^{-1}(u)\) is:

\[
\frac{d}{dx} \left[ \sin^{-1}(u) \right] = \frac{1}{\sqrt{1 - u^2}}
\]

So:

\[
f'(x) = \frac{1}{\sqrt{1 - (14 \cos(x))^2}} \cdot \frac{d}{dx} \left[ 14 \cos(x) \right]
\]

### Step 2: Differentiate the inner function.

The derivative of \(14 \cos(x)\) with respect to \(x\) is:

\[
\frac{d}{dx} \left[ 14 \cos(x) \right] = -14 \sin(x)
\]

### Step 3: Combine the results.

Now substitute this into the derivative expression:

\[
f'(x) = \frac{-14 \sin(x)}{\sqrt{1 - (14 \cos(x))^2}}
\]

### Step 4: Simplify the denominator.

We simplify the expression in the square root:

\[
(14 \cos(x))^2 = 196 \cos^2(x)
\]

Thus, the final expression for the derivative is:

\[
f'(x) = \frac{-14 \sin(x)}{\sqrt{1 - 196 \cos^2(x)}}
\]

This is the derivative of the function.

Would you like a more detailed explanation on any part? Here are some related questions:

1. How do you apply the chain rule to composite functions?
2. Why is the derivative of \(\sin^{-1}(x)\) \(\frac{1}{\sqrt{1 - x^2}}\)?
3. Can you explain how to differentiate trigonometric functions like \(\cos(x)\)?
4. How does the domain of the inverse sine function affect the solution?
5. How can we check if the solution is correct?

**Tip**: When dealing with inverse trigonometric functions, it's helpful to remember the derivative formulas for each and how to apply the chain rule properly.

Find the derivative of the function f(x) = sin^(-1)(14 cos x).

Learn how to compute the derivative of f(x) = sin^(-1)(14 cos(x)) using the chain rule and inverse trigonometric derivative formulas. This problem involves applying calculus concepts such as the chain rule and differentiation of inverse sine functions.

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