AP Calculus BC 2.5 Applying the Power Rule Study Notes - New Syllabus
AP Calculus BC 2.5 Applying the Power Rule Study Notes- New syllabus
AP Calculus BC 2.5 Applying the Power Rule Study Notes – AP Calculus BC- per latest AP Calculus BC Syllabus.
LEARNING OBJECTIVE
- Recognizing opportunities to apply derivative rules can simplify differentiation.
Key Concepts:
- Power Rule
Applying the Power Rule
Applying the Power Rule
The Power Rule helps us differentiate any function of the form \( f(x) = x^n \), where \( n \) is any real number positive, negative, or fractional.
Power Rule Formula:
If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \)
This rule works for:
- Positive integers (e.g., \( x^3 \))
- Negative exponents (e.g., \( x^{-2} \))
- Fractional exponents (e.g., \( x^{1/2} \))
Example :
Differentiate \( f(x) = 7x^5 – 3x^2 + 4 \)
▶️Answer/Explanation
- \( \frac{d}{dx}(7x^5) = 7 \cdot 5x^{4} = 35x^4 \)
- \( \frac{d}{dx}(-3x^2) = -6x \)
- \( \frac{d}{dx}(4) = 0 \)
Final Answer: \( f'(x) = 35x^4 – 6x \)
Example:
Differentiate \( f(x) = 5x^{-3} \)
▶️Answer/Explanation
- Apply Power Rule: \( f'(x) = 5 \cdot (-3)x^{-4} = -15x^{-4} \)
Final Answer: \( f'(x) = -15x^{-4} \)
Example:
Differentiate \( f(x) = 3x^{1/2} \)
▶️Answer/Explanation
- Apply Power Rule: \( f'(x) = 3 \cdot \frac{1}{2}x^{-1/2} = \frac{3}{2\sqrt{x}} \)
Final Answer: \( f'(x) = \frac{3}{2\sqrt{x}} \)
Example :
Differentiate \( f(x) = 4x^3 – 2x^{-2} + 5x^{1/2} \)
▶️Answer/Explanation
- \( \frac{d}{dx}(4x^3) = 4 \cdot 3x^2 = 12x^2 \)
- \( \frac{d}{dx}(-2x^{-2}) = -2 \cdot (-2)x^{-3} = 4x^{-3} \)
- \( \frac{d}{dx}(5x^{1/2}) = 5 \cdot \frac{1}{2}x^{-1/2} = \frac{5}{2\sqrt{x}} \)
Final Answer: \( f'(x) = 12x^2 + 4x^{-3} + \frac{5}{2\sqrt{x}} \)