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.

Learn how to compute the derivative of the function x^4 h^4 + y^3 + z^5 with respect to h. This solution applies the power rule of differentiation and the constant rule. Suitable for students in Grades 11-12 studying calculus.

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