https://file.aiquickdraw.com/mathbot/file/1730128363344lzoruqlx.jpg

Let's solve each part of this problem by calculating the derivatives and observing any patterns.

### Part (a): \( f(x) = x^n \)
1. **First derivative**:  
   \[
   f'(x) = n \cdot x^{n-1}
   \]

2. **Second derivative**:  
   \[
   f''(x) = n(n-1) \cdot x^{n-2}
   \]

3. **Third derivative**:  
   \[
   f'''(x) = n(n-1)(n-2) \cdot x^{n-3}
   \]

4. **General form**:  
   Observing the pattern, we see that the \( k \)-th derivative will be:
   \[
   f^{(k)}(x) = n(n-1)(n-2) \dots (n-k+1) \cdot x^{n-k}
   \]

5. **Nth derivative**:  
   When \( k = n \), we get:
   \[
   f^{(n)}(x) = n! \quad \text{(if } n \geq 0\text{)}
   \]
   For \( k > n \), all derivatives will be zero because we will eventually reach a constant term.

### Part (b): \( f(x) = \frac{1}{x} = x^{-1} \)
1. **First derivative**:  
   \[
   f'(x) = -x^{-2} = -\frac{1}{x^2}
   \]

2. **Second derivative**:  
   \[
   f''(x) = 2x^{-3} = \frac{2}{x^3}
   \]

3. **Third derivative**:  
   \[
   f'''(x) = -6x^{-4} = -\frac{6}{x^4}
   \]

4. **General pattern**:  
   It appears that the \( n \)-th derivative is:
   \[
   f^{(n)}(x) = (-1)^n \cdot n! \cdot x^{-(n+1)} = (-1)^n \frac{n!}{x^{n+1}}
   \]

Let me know if you'd like a more detailed breakdown of these steps or if you have any questions.

---

**Follow-up Questions:**
1. What is the significance of the factorials in these derivative patterns?
2. How can we find patterns in derivatives for other functions like \( e^x \) or \( \sin(x) \)?
3. How does the power rule work in higher-order derivatives for any power \( n \)?
4. What happens to the derivative pattern when \( n \) is a negative or non-integer?
5. Can we use similar methods to find higher-order derivatives of logarithmic functions?

**Tip:** To check patterns in derivatives, write out the first few terms to spot relationships between the coefficients and exponents.

Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs. (a) f(x) = x^n (b) f(x) = 1/x

Learn how to find the nth derivative of the functions f(x) = x^n and f(x) = 1/x by calculating initial derivatives and observing patterns. This solution includes the use of the power rule and factorials, suitable for college-level calculus.

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