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To solve the problem, we need to identify the inner function \( g(x) \) and the outer function \( f(u) \) such that \( y = f(g(x)) \), and then find the derivative \(\frac{dy}{dx}\).

Given:
\[ y = \tan(x^5) \]

### Step 1: Identify the Inner and Outer Functions

Let's set the inner function \( g(x) \) and the outer function \( f(u) \):
- Inner function: \( g(x) = x^5 \)
- Outer function: \( f(u) = \tan(u) \)

Thus, we can write:
\[ y = f(g(x)) = \tan(x^5) \]

### Step 2: Compute the Derivative

To find the derivative \(\frac{dy}{dx}\), we use the chain rule:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]

First, compute \( g'(x) \):
\[ g(x) = x^5 \]
\[ g'(x) = 5x^4 \]

Next, compute \( f'(u) \):
\[ f(u) = \tan(u) \]
\[ f'(u) = \sec^2(u) \]

Since \( u = g(x) = x^5 \), we have:
\[ f'(g(x)) = \sec^2(x^5) \]

Therefore,
\[ \frac{dy}{dx} = \sec^2(x^5) \cdot 5x^4 \]

### Final Answer

\[ \frac{dy}{dx} = 5x^4 \sec^2(x^5) \]

Would you like more details or have any questions about this solution? Here are five related questions to consider:

1. What is the derivative of \( y = \tan(3x^2) \)?
2. How do you apply the chain rule for the function \( y = \sin(\ln(x)) \)?
3. What is the inner function for \( y = \cos(e^x) \)?
4. How can you find the derivative of a composite function using implicit differentiation?
5. What is the second derivative of \( y = \tan(x^5) \)?

**Tip:** When dealing with composite functions, always identify the inner and outer functions clearly before applying the chain rule.

Write the composite function in the form f(g(x)). Identify the inner function u = g(x) and the outer function y = f(u). Given: y = tan(x^5).

Learn how to find the derivative of y = tan(x^5) by identifying the inner and outer functions and applying the chain rule. This step-by-step solution covers key calculus concepts and is suitable for students in Grades 11-12 or introductory college calculus.

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