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To solve for the derivative of \( \tan^{-1} \left( e^{5 \cos x} \right) \), we will use the chain rule. Here's how the process works step by step:

### Step 1: Recall the derivative of \( \tan^{-1}(u) \)
The derivative of \( \tan^{-1}(u) \) with respect to \( u \) is:
\[
\frac{d}{du} \left( \tan^{-1}(u) \right) = \frac{1}{1 + u^2}
\]
In this case, \( u = e^{5 \cos x} \).

### Step 2: Apply the chain rule
We need to differentiate \( \tan^{-1} \left( e^{5 \cos x} \right) \) with respect to \( x \). According to the chain rule:
\[
\frac{d}{dx} \left( \tan^{-1}(u) \right) = \frac{1}{1 + u^2} \cdot \frac{du}{dx}
\]
Here, \( u = e^{5 \cos x} \), so we first need to find \( \frac{du}{dx} \).

### Step 3: Differentiate \( e^{5 \cos x} \)
To find \( \frac{du}{dx} \), we differentiate \( e^{5 \cos x} \) using the chain rule. First, the derivative of \( e^{g(x)} \) is \( e^{g(x)} \cdot g'(x) \), where \( g(x) = 5 \cos x \).

So,
\[
\frac{d}{dx} \left( e^{5 \cos x} \right) = e^{5 \cos x} \cdot (-5 \sin x)
\]
Thus,
\[
\frac{du}{dx} = e^{5 \cos x} \cdot (-5 \sin x)
\]

### Step 4: Substitute everything back into the chain rule
Now substitute \( u = e^{5 \cos x} \) and \( \frac{du}{dx} = e^{5 \cos x} \cdot (-5 \sin x) \) into the chain rule expression:
\[
\frac{d}{dx} \left( \tan^{-1} \left( e^{5 \cos x} \right) \right) = \frac{1}{1 + \left( e^{5 \cos x} \right)^2} \cdot \left( e^{5 \cos x} \cdot (-5 \sin x) \right)
\]

### Final Answer:
The derivative is:
\[
\frac{-5 e^{5 \cos x} \sin x}{1 + \left( e^{5 \cos x} \right)^2}
\]

Would you like any additional details or clarifications on this process?

### Related Questions:
1. What is the derivative of \( \tan^{-1}(x^2) \)?
2. How would you differentiate \( \tan^{-1}(\sin x) \)?
3. What is the derivative of \( e^{\sin x} \)?
4. How do you find the derivative of inverse trigonometric functions in general?
5. What role does the chain rule play in differentiating composite functions?

### Tip:
Always remember to apply the chain rule when differentiating composite functions, especially with inverse trigonometric and exponential expressions.

Select the correct derivative of tan^(-1)(e^(5cos(x)))

Learn how to find the derivative of tan^(-1)(e^(5cos(x))) using the chain rule. This problem explores inverse trigonometric functions and exponential derivatives, ideal for advanced calculus students.

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