f(x) = \tan^{-1}(3^x) find the derivative

To find the derivative of the function \( f(x) = \tan^{-1}(3^x) \), we will use the chain rule. Let's go through the steps:

1. **Identify the outer function**: The outer function is \( \tan^{-1}(u) \), where \( u = 3^x \).
2. **Identify the inner function**: The inner function is \( 3^x \).

The chain rule states that if \( f(x) = g(h(x)) \), then \( f'(x) = g'(h(x)) \cdot h'(x) \).

### Step-by-Step Solution

1. **Differentiate the outer function** \( g(u) = \tan^{-1}(u) \):
   The derivative of \( \tan^{-1}(u) \) with respect to \( u \) is:
   \[
   \frac{d}{du} \left( \tan^{-1}(u) \right) = \frac{1}{1 + u^2}
   \]

2. **Differentiate the inner function** \( h(x) = 3^x \):
   The derivative of \( 3^x \) with respect to \( x \) is:
   \[
   \frac{d}{dx} \left( 3^x \right) = 3^x \ln(3)
   \]

3. **Apply the chain rule**:
   Combining these results using the chain rule, we get:
   \[
   f'(x) = \frac{d}{dx} \left( \tan^{-1}(3^x) \right) = \frac{1}{1 + (3^x)^2} \cdot 3^x \ln(3)
   \]

4. **Simplify the expression**:
   \[
   f'(x) = \frac{3^x \ln(3)}{1 + 9^x}
   \]

Therefore, the derivative of the function \( f(x) = \tan^{-1}(3^x) \) is:
\[
f'(x) = \frac{3^x \ln(3)}{1 + 9^x}
\]

Would you like more details or have any questions? Here are five questions you might consider:

1. How can we use the derivative \( f'(x) \) to analyze the behavior of the function \( f(x) = \tan^{-1}(3^x) \)?
2. What is the second derivative of the function \( f(x) = \tan^{-1}(3^x) \)?
3. Can you explain the steps of the chain rule in more detail?
4. How does the derivative \( f'(x) \) help in finding critical points of the function \( f(x) = \tan^{-1}(3^x) \)?
5. What are the applications of the derivative \( f'(x) = \frac{3^x \ln(3)}{1 + 9^x} \) in real-world problems?

**Tip:** When applying the chain rule, carefully identify the outer and inner functions and their respective derivatives. This will simplify the process and help avoid mistakes.

Learn how to find the derivative of the function arctan(3^x) step-by-step using calculus and the chain rule. Understand the application of derivatives in analyzing trigonometric and exponential functions.

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