Find the first partial derivatives of the function.

*f*(*r*, *s*) = *r* ln(*r*5 + *s*4)

We are given the function:

\[
f(r, s) = r \ln(r^5 + s^4)
\]

We need to find the first partial derivatives of \( f(r, s) \) with respect to \( r \) and \( s \).

### 1. Partial derivative of \( f(r, s) \) with respect to \( r \):

Using the product rule and chain rule, we first differentiate the outer function \( f(r, s) = r \ln(g(r, s)) \) where \( g(r, s) = r^5 + s^4 \).

\[
\frac{\partial f}{\partial r} = \frac{\partial}{\partial r} \left( r \ln(r^5 + s^4) \right)
\]

Applying the product rule:
\[
\frac{\partial f}{\partial r} = \ln(r^5 + s^4) + r \cdot \frac{1}{r^5 + s^4} \cdot \frac{\partial}{\partial r}(r^5 + s^4)
\]

Now, compute the derivative of \( r^5 + s^4 \) with respect to \( r \):
\[
\frac{\partial}{\partial r}(r^5 + s^4) = 5r^4
\]

Thus, we have:
\[
\frac{\partial f}{\partial r} = \ln(r^5 + s^4) + \frac{r \cdot 5r^4}{r^5 + s^4}
\]

Simplifying:
\[
\frac{\partial f}{\partial r} = \ln(r^5 + s^4) + \frac{5r^5}{r^5 + s^4}
\]

### 2. Partial derivative of \( f(r, s) \) with respect to \( s \):

Now, we differentiate \( f(r, s) = r \ln(r^5 + s^4) \) with respect to \( s \), applying the chain rule.

\[
\frac{\partial f}{\partial s} = r \cdot \frac{1}{r^5 + s^4} \cdot \frac{\partial}{\partial s}(r^5 + s^4)
\]

Now, compute the derivative of \( r^5 + s^4 \) with respect to \( s \):
\[
\frac{\partial}{\partial s}(r^5 + s^4) = 4s^3
\]

Thus, we have:
\[
\frac{\partial f}{\partial s} = r \cdot \frac{4s^3}{r^5 + s^4}
\]

### Final Results:
\[
\frac{\partial f}{\partial r} = \ln(r^5 + s^4) + \frac{5r^5}{r^5 + s^4}
\]

\[
\frac{\partial f}{\partial s} = \frac{4rs^3}{r^5 + s^4}
\]

Would you like more details on any specific part of this process?

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Here are some related questions:

1. How does the product rule apply in this case?
2. Why do we use the chain rule for the logarithmic term?
3. Can we further simplify the partial derivatives?
4. How would you compute the second partial derivatives?
5. What is the significance of the mixed partial derivatives in this context?

**Tip:** Always remember to simplify expressions after differentiation to make further analysis easier.

Find the first partial derivatives of the function f(r, s) = r ln(r^5 + s^4).

Explore how to find the first partial derivatives of the function f(r, s) = r ln(r^5 + s^4) by applying the chain rule and product rule. The detailed step-by-step solution covers advanced calculus concepts, including logarithmic functions and partial derivatives, suitable for university-level students.

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