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The given function is \( n(t) = 3 \sqrt{t^2} - \frac{5}{t^{3/2}} \).

To differentiate this function, we will follow these steps:

1. **Simplify the function (if possible)**:
   - \(\sqrt{t^2} = t\), since \( t \) is typically taken to be non-negative in this context.
   - Rewrite the second term \(\frac{5}{t^{3/2}}\) using negative exponents: \( \frac{5}{t^{3/2}} = 5t^{-3/2} \).

Thus, the function becomes:
\[ n(t) = 3t - 5t^{-3/2} \]

2. **Differentiate each term separately**:
   - For the term \( 3t \): Use the power rule, \(\frac{d}{dt}(t^n) = nt^{n-1}\).
     \[
     \frac{d}{dt}(3t) = 3 \cdot 1 t^{1-1} = 3
     \]
   - For the term \( -5t^{-3/2} \): Use the power rule.
     \[
     \frac{d}{dt}(-5t^{-3/2}) = -5 \cdot \left( -\frac{3}{2} \right) t^{-3/2 - 1} = \frac{15}{2} t^{-5/2}
     \]

3. **Combine the differentiated terms**: Add the derivatives of each term to get the derivative of the entire function.
   \[
   n'(t) = 3 + \frac{15}{2} t^{-5/2}
   \]

So, the derivative of \( n(t) = 3t - 5t^{-3/2} \) is \( n'(t) = 3 + \frac{15}{2} t^{-5/2} \).

Would you like more details or have any questions?

Here are 8 related questions for further exploration:
1. How do you find the second derivative of \( n(t) = 3t - 5t^{-3/2} \)?
2. What is the derivative of the function \( m(t) = t^3 - \frac{7}{t^2} \)?
3. How do you apply the product rule to the function \( p(t) = t^2 \cdot e^t \)?
4. What are the critical points of the function \( n(t) = 3t - 5t^{-3/2} \)?
5. How can you use differentiation to find the local maxima and minima of a function?
6. What is the derivative of the logarithmic function \( \ln(t) \)?
7. How do you differentiate the trigonometric function \( \cos(t) \)?
8. What are the applications of differentiation in physics problems, such as finding velocity and acceleration?

**Tip:** When dealing with functions involving roots or fractional exponents, it can be helpful to rewrite them using negative exponents for easier differentiation.

Understand how to apply the differentiation rule step-by-step with examples. Learn about the power rule, handling negative exponents, and practical applications. Suitable for advanced high school students and college-level learners.

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