AP Calculus BC 2.6 Derivative Rules Study Notes - New Syllabus
AP Calculus BC 2.6 Derivative Rules Study Notes- New syllabus
AP Calculus BC 2.6 Derivative Rules Study Notes – AP Calculus BC- per latest AP Calculus BC Syllabus.
LEARNING OBJECTIVE
- Sums, differences, and constant multiples of functions can be differentiated using derivative rules.
Key Concepts:
- Derivative Rules: Constant, Sum, Difference, and Constant Multiple
Derivative Rules: Constant, Sum, Difference, and Constant Multiple
Constant Rule
The derivative of any constant value is always zero.
Formula: If \( f(x) = c \), where \( c \) is a constant, then \( f'(x) = 0 \)
Example:
Differentiate \( f(x) = 12 \)
▶️Answer/Explanation
Since it’s a constant, \( f'(x) = 0 \)
Example:
Differentiate \( f(x) = -7 \)
▶️Answer/Explanation
Again, it’s constant, so \( f'(x) = 0 \)
Sum Rule
The derivative of the sum of two functions is the sum of their derivatives.
Formula: If \( f(x) = u(x) + v(x) \), then \( f'(x) = u'(x) + v'(x) \)
Example:
Differentiate \( f(x) = x^2 + 3x \)
▶️Answer/Explanation
- \( \frac{d}{dx}(x^2) = 2x \)
- \( \frac{d}{dx}(3x) = 3 \)
Final Answer: \( f'(x) = 2x + 3 \)
Example:
Differentiate \( f(x) = x^3 + x^5 \)
▶️Answer/Explanation
- \( \frac{d}{dx}(x^3) = 3x^2 \)
- \( \frac{d}{dx}(x^5) = 5x^4 \)
Final Answer: \( f'(x) = 3x^2 + 5x^4 \)
Difference Rule
The derivative of the difference of two functions is the difference of their derivatives.
Formula: If \( f(x) = u(x) – v(x) \), then \( f'(x) = u'(x) – v'(x) \)
Example:
Differentiate \( f(x) = x^4 – 2x^2 \)
▶️Answer/Explanation
- \( \frac{d}{dx}(x^4) = 4x^3 \)
- \( \frac{d}{dx}(2x^2) = 4x \)
Final Answer: \( f'(x) = 4x^3 – 4x \)
Example:
Differentiate \( f(x) = 5x^3 – x \)
▶️Answer/Explanation
- \( \frac{d}{dx}(5x^3) = 15x^2 \)
- \( \frac{d}{dx}(x) = 1 \)
Final Answer: \( f'(x) = 15x^2 – 1 \)
Constant Multiple Rule
The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.
Formula: If \( f(x) = c \cdot u(x) \), then \( f'(x) = c \cdot u'(x) \)
Example :
Differentiate \( f(x) = 7x^3 \)
▶️Answer/Explanation
\( f'(x) = 7 \cdot \frac{d}{dx}(x^3) = 7 \cdot 3x^2 = 21x^2 \)
Example :
Differentiate \( f(x) = -4x^{1/2} \)
▶️Answer/Explanation
\( f'(x) = -4 \cdot \frac{1}{2}x^{-1/2} = -\frac{2}{\sqrt{x}} \)
| Rule | Formula | Description |
|---|---|---|
| Constant Rule | \( \frac{d}{dx}(c) = 0 \) | The derivative of any constant is 0. |
| Sum Rule | \( \frac{d}{dx}[u + v] = \frac{du}{dx} + \frac{dv}{dx} \) | The derivative of a sum is the sum of derivatives. |
| Difference Rule | \( \frac{d}{dx}[u – v] = \frac{du}{dx} – \frac{dv}{dx} \) | The derivative of a difference is the difference of derivatives. |
| Constant Multiple Rule | \( \frac{d}{dx}[c \cdot u] = c \cdot \frac{du}{dx} \) | A constant can be pulled outside the derivative. |