AP Calculus BC 2.9 The Quotient Rule Study Notes - New Syllabus
AP Calculus BC 2.9 The Quotient Rule Study Notes- New syllabus
AP Calculus BC 2.9 The Quotient Rule Study Notes – AP Calculus BC- per latest AP Calculus BC Syllabus.
LEARNING OBJECTIVE
- Calculate derivatives of quotients of differentiable functions.
Key Concepts:
- Quotient Rule
Quotient Rule
Quotient Rule
The Quotient Rule is used when you’re differentiating a function that is the division of two functions.
Formula: If \( f(x) = \dfrac{u(x)}{v(x)} \), then:
\( f'(x) = \dfrac{u'(x)v(x) – u(x)v'(x)}{[v(x)]^2} \)
Mnemonic: “Low D High minus High D Low over Low squared”
- Low = denominator
- High = numerator
- D = derivative
Example:
Differentiate \( f(x) = \dfrac{x^2}{x + 1} \)
▶️Answer/Explanation
Let \( u(x) = x^2 \), \( v(x) = x + 1 \)
- \( u'(x) = 2x \)
- \( v'(x) = 1 \)
Apply Quotient Rule:
\( f'(x) = \dfrac{2x(x + 1) – x^2(1)}{(x + 1)^2} \)
Simplify numerator:
\( f'(x) = \dfrac{2x^2 + 2x – x^2}{(x + 1)^2} = \dfrac{x^2 + 2x}{(x + 1)^2} \)
Example:
Differentiate \( f(x) = \dfrac{\sin x}{x^2} \)
▶️Answer/Explanation
Let \( u(x) = \sin x \), \( v(x) = x^2 \)
- \( u'(x) = \cos x \)
- \( v'(x) = 2x \)
Apply Quotient Rule:
\( f'(x) = \dfrac{\cos x \cdot x^2 – \sin x \cdot 2x}{x^4} \)
Final Answer: \( f'(x) = \dfrac{x^2 \cos x – 2x \sin x}{x^4} \)
Example:
Differentiate \( f(x) = \dfrac{\ln x}{x} \)
▶️Answer/Explanation
Let \( u(x) = \ln x \), \( v(x) = x \)
- \( u'(x) = \frac{1}{x} \)
- \( v'(x) = 1 \)
Apply Quotient Rule:
\( f'(x) = \dfrac{\frac{1}{x} \cdot x – \ln x \cdot 1}{x^2} = \dfrac{1 – \ln x}{x^2} \)
Example:
Differentiate \( f(x) = \dfrac{e^x}{\cos x} \)
▶️Answer/Explanation
Let \( u(x) = e^x \), \( v(x) = \cos x \)
- \( u'(x) = e^x \)
- \( v'(x) = -\sin x \)
Apply Quotient Rule:
\( f'(x) = \dfrac{e^x \cdot \cos x – e^x \cdot (-\sin x)}{(\cos x)^2} \)
Simplify:
\( f'(x) = \dfrac{e^x(\cos x + \sin x)}{\cos^2 x} \)