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We are tasked with finding the third derivative \( \frac{d^3 y}{dx^3} \) for the given function:

\[
y = \sqrt{x} \ln{x} + x^4 + \pi^3
\]

### Step-by-step approach:

1. **First Derivative \( \frac{dy}{dx} \)**:
   We will use the product rule for the first term \( \sqrt{x} \ln{x} = x^{1/2} \ln{x} \) and standard derivative rules for the rest.

2. **Second Derivative \( \frac{d^2 y}{dx^2} \)**:
   Differentiate the result from the first step.

3. **Third Derivative \( \frac{d^3 y}{dx^3} \)**:
   Finally, differentiate the second derivative to get the required third derivative.

Let me compute this now.Here are the results of the derivatives:

1. **First derivative \( \frac{dy}{dx} \)**:
   \[
   \frac{dy}{dx} = 4x^3 + \frac{\ln{x}}{2\sqrt{x}} + \frac{1}{\sqrt{x}}
   \]

2. **Second derivative \( \frac{d^2y}{dx^2} \)**:
   \[
   \frac{d^2y}{dx^2} = 12x^2 - \frac{\ln{x}}{4x^{3/2}}
   \]

3. **Third derivative \( \frac{d^3y}{dx^3} \)**:
   \[
   \frac{d^3y}{dx^3} = 24x + \frac{3\ln{x}}{8x^{5/2}} - \frac{1}{4x^{5/2}}
   \]

Would you like further explanation of any of the steps or details on any specific part of the derivatives?

### Here are 5 related questions to explore:
1. How do you apply the product rule when differentiating functions like \( \sqrt{x} \ln{x} \)?
2. What are the general rules for differentiating logarithmic and power functions?
3. How do higher-order derivatives (second, third, etc.) behave as functions become more complex?
4. Why does the constant \( \pi^3 \) disappear after the first derivative?
5. What is the physical significance of third-order derivatives in certain real-world contexts (like acceleration)?

**Tip:** When dealing with functions that involve both logarithms and powers of \(x\), using the product and chain rules carefully will ensure accuracy in derivatives.

Determine the third derivative \( \frac{d^3 y}{dx^3} \) where \( y = \sqrt{x} \ln{x} + x^4 + \pi^3 \).

Learn how to find the third derivative of the function \( y = \sqrt{x} \ln{x} + x^4 + \pi^3 \) using the product rule, power rule, and logarithmic differentiation. This problem is ideal for college calculus students looking to understand higher-order derivatives.

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