To find the derivative of $x^4 h^4 + y^3 + z^5$ with respect to $h$, we'll differentiate each term individually.

1. $x^4 h^4$: Treat $x$ as a constant because we are differentiating with respect to $h$. The derivative of $h^4$ is $4h^3$, so: $\frac{d}{dh}(x^4 h^4) = x^4 \cdot 4h^3 = 4x^4 h^3$

2. $y^3$: Since $y$ is independent of $h$, the derivative of this term with respect to $h$ is 0: $\frac{d}{dh}(y^3) = 0$

3. $z^5$: Similarly, since $z$ is independent of $h$, the derivative of this term with respect to $h$ is 0: $\frac{d}{dh}(z^5) = 0$

Now, putting it all together: $\frac{d}{dh}(x^4 h^4 + y^3 + z^5) = 4x^4 h^3$

Thus, the derivative is: $\boxed{4x^4 h^3}$

Would you like more details on this, or do you have any questions?

Here are 5 related questions to consider:

1. What is the derivative of $x^n h^m$ with respect to $h$?
2. How do we apply the product rule when differentiating multiple variables?
3. What is the second derivative of $x^4 h^4 + y^3 + z^5$ with respect to $h$?
4. How does differentiating implicitly with respect to $h$ change when all variables are functions of $h$?
5. What are partial derivatives, and how do they differ from regular derivatives?

Tip: When differentiating terms involving multiple variables, always determine which variable you are differentiating with respect to and treat the others as constants unless otherwise specified.