Edexcel IAL - Pure Maths 1- 4.2 Differentiation of xn and Related Expressions- Study notes - New syllabus
Edexcel IAL – Pure Maths 1- 4.2 Differentiation of xn and Related Expressions -Study notes- New syllabus
Edexcel IAL – Pure Maths 1- 4.2 Differentiation of xn and Related Expressions -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- Differentiation of xn and Related Expressions
Differentiation of \( x^n \) and Related Rules
The Power Rule
If \( n \) is any real number, the derivative of \( x^n \) is:
\( \dfrac{d}{dx}(x^n) = nx^{\,n-1} \)
This rule works for positive integers, negative powers, and fractional powers.
Related Differentiation Rules
| Rule | Formula |
| Constant Multiple Rule | \( \dfrac{d}{dx}(k f(x)) = k f'(x) \) |
| Sum Rule | \( \dfrac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) \) |
| Difference Rule | \( \dfrac{d}{dx}[f(x) – g(x)] = f'(x) – g'(x) \) |
Key Notes
- Each term is differentiated separately.
- Constants multiply through but disappear when differentiating alone.
- Always rewrite expressions with powers before differentiating.
Example
Differentiate \( y = 7x^3 \).
▶️ Answer / Explanation
Use the power rule:
\( y’ = 7 \cdot 3x^2 = 21x^2 \)
Example
Differentiate \( y = 5x^4 – 3x^2 + 9 \).
▶️ Answer / Explanation
Differentiate term by term:
\( y’ = 20x^3 – 6x \)
Example
Differentiate \( y = \dfrac{4}{\sqrt{x}} – 3x^{-2} + 7x^{\,5/2}. \)
▶️ Answer / Explanation
Rewrite with powers:
\( y = 4x^{-1/2} – 3x^{-2} + 7x^{5/2} \)