IB Mathematics AA Derivative of f(x) Study Notes
IB Mathematics AA Derivative of f(x) Study Notes
IB Mathematics AA Derivative of f(x) Notes Offer a clear explanation of Use of Derivative of f(x), including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Derivative of f(x).
Derivative of Power Functions
Derivative of Power Functions
If \( f(x) = ax^n \) where \( a \) is a constant and \( n \in \mathbb{Z} \) (an integer), then the derivative is:
\( f'(x) = a n x^{n – 1} \)
This rule applies to positive, negative, and zero integer powers.
Example:
Find the derivative of \( f(x) = 5x^4 \).
▶️ Answer/Explanation
Apply the power rule
For \( 5x^4 \):
\( \frac{d}{dx} 5x^4 = 5 \cdot 4 x^{3} = 20x^3 \)
Derivative of Functions of the Form \(ax^n + bx^{n-1} +\cdots \)
Derivative of Functions of the Form \(ax^n + bx^{n-1} +\cdots \)
For functions like:
\( f(x) = ax^n + bx^{n-1} + cx^{n-2} + \cdots \) where all exponents are integers, the derivative is found by applying the power rule to each term:
\( f'(x) = a n x^{n-1} + b (n-1) x^{n-2} + c (n-2) x^{n-3} + \cdots \)
The derivative is the sum of the derivatives of each term.
Example:
Differentiate \( f(x) = 2x^5 – 3x^3 + 4x^{-1} – 7 \).
▶️ Answer/Explanation
Apply the power rule term-by-term
- \( \frac{d}{dx}(2x^5) = 2 \cdot 5 x^4 = 10x^4 \)
- \( \frac{d}{dx}(-3x^3) = -3 \cdot 3 x^2 = -9x^2 \)
- \( \frac{d}{dx}(4x^{-1}) = 4 \cdot (-1) x^{-2} = -4x^{-2} \)
- \( \frac{d}{dx}(-7) = 0 \)
Final derivative:
\( f'(x) = 10x^4 – 9x^2 – 4x^{-2} \)