#Power Rule and Differentiation

#Table of Contents

1. Introduction to the Power Rule
2. Differentiating Powers of x
   • Basic Differentiation
   • Examples
   • Fractional and Negative Powers
3. Worked Example
4. Derivatives of Sums, Differences, and Constant Multiples
   • Sums and Differences
   • Constant Multiples
   • Special Cases
5. Simplifying Expressions to Find Derivatives
   • Expansion
   • Using Laws of Exponents
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways

#1. Introduction to the Power Rule

The power rule is a fundamental tool in calculus for differentiating functions of the form $x^n$. It simplifies the process of finding the derivative of polynomials and other functions involving powers of $x$.

#2. Differentiating Powers of x

#Basic Differentiation

#Examples

1. Simple Power:

   • If $f(x) = x^7$, then: $$f'(x) = 7x^{6}$$

2. Fractional Power:

   • If $g(x) = x^{\frac{2}{3}}$, then: $$g'(x) = \frac{2}{3} x^{-\frac{1}{3}}$$

3. **Negati...