We are given the function $h(x) = 4^{x^9}$ and are tasked with finding its first derivative, $h'(x)$.

To solve this, we'll apply the chain rule and logarithmic differentiation. Here’s the step-by-step solution:

Step 1: Rewrite the function using logarithms

We know that $a^b$ can be written as $e^{b \ln a}$. So, we can express $h(x) = 4^{x^9}$ as: $h(x) = e^{x^9 \ln 4}$

Step 2: Differentiate using the chain rule

Now, differentiate $h(x) = e^{x^9 \ln 4}$ with respect to $x$: $h'(x) = \frac{d}{dx} \left( e^{x^9 \ln 4} \right)$ The derivative of $e^u$ is $e^u \cdot \frac{du}{dx}$, where $u = x^9 \ln 4$. The derivative of $u$ is: $\frac{du}{dx} = 9x^8 \ln 4$ Thus, the derivative becomes: $h'(x) = e^{x^9 \ln 4} \cdot 9x^8 \ln 4$

Step 3: Substitute back $e^{x^9 \ln 4} = 4^{x^9}$

Substituting $e^{x^9 \ln 4} = 4^{x^9}$ back into the expression, we get: $h'(x) = 4^{x^9} \cdot 9x^8 \ln 4$

Step 4: Identify the correct answer

The correct derivative is: $h'(x) = 9x^8 \cdot 4^{x^9} \ln 4$

Looking at the options, the correct answer is:

$\boxed{h'(x) = (9x^8) 4^{x^9} \ln 4}$

Let me know if you need more details or further explanations.

Here are 5 related questions:

1. How does the chain rule apply to exponential functions?
2. What is the logarithmic differentiation method, and why is it useful?
3. How can we generalize the differentiation of $a^{f(x)}$ for any base $a$?
4. What are some real-world applications of differentiating exponential functions?
5. How do we interpret the derivative of $4^{x^9}$ geometrically?

Tip: When differentiating exponential functions, rewriting them using natural logarithms simplifies the process significantly.