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.