The differential of \( x^n \) with respect to \( x \) is calculated using the power rule of differentiation, which states:

\[
\frac{d}{dx} \left( x^n \right) = n x^{n-1}
\]

### Explanation:
- \( n \) is a constant exponent.
- The derivative of \( x^n \) brings the exponent \( n \) down as a coefficient, and then subtracts 1 from the exponent.

#### Example:
If \( f(x) = x^5 \), then its derivative is:
\[
\frac{d}{dx}(x^5) = 5x^{4}
\]

Would you like more details or have any questions about this process?

### Related questions:
1. What is the derivative of \( x^n \) when \( n \) is negative?
2. How do you apply the power rule to functions like \( (2x)^n \)?
3. What happens when \( n = 0 \) in \( x^n \)?
4. Can you differentiate \( x^n \) multiple times (higher-order derivatives)?
5. How does the power rule apply to fractional exponents?

**Tip:** The power rule is one of the most fundamental differentiation techniques in calculus—memorizing it will speed up solving many calculus problems.

Understand how to differentiate x^n using the power rule in calculus. This solution explains the formula d/dx(x^n) = nx^(n-1) with examples and detailed steps. Suitable for high school and college-level students learning calculus.

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