Derivative Quotient Rule
Flashcard
•
Mathematics
•
12th Grade
•
Practice Problem
•
Hard
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1.
FLASHCARD QUESTION
Front
What is the Quotient Rule in calculus?
Back
The Quotient Rule states that if you have a function that is the quotient of two functions, say \( f(x) = \frac{g(x)}{h(x)} \), then the derivative is given by \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \).
2.
FLASHCARD QUESTION
Front
Differentiate the function \( y = \frac{x^2 + 6}{2x - 7} \).
Back
Using the Quotient Rule, \( y' = \frac{(2x)(2x - 7) - (x^2 + 6)(2)}{(2x - 7)^2} = \frac{2x^2 - 14x - 12}{(2x - 7)^2} \).
3.
FLASHCARD QUESTION
Front
Find the derivative of \( f(x) = \frac{2x - 7}{x + 3} \).
Back
Using the Quotient Rule, \( f'(x) = \frac{(2)(x + 3) - (2x - 7)(1)}{(x + 3)^2} = \frac{13}{(x + 3)^2} \).
4.
FLASHCARD QUESTION
Front
What is the first step in applying the Quotient Rule?
Back
Identify the numerator and denominator functions, \( g(x) \) and \( h(x) \), respectively, in the quotient \( f(x) = \frac{g(x)}{h(x)} \).
5.
FLASHCARD QUESTION
Front
Differentiate \( y = \frac{2}{2x^4 - 5} \).
Back
Using the Quotient Rule, \( y' = \frac{0(2x^4 - 5) - 2(8x^3)}{(2x^4 - 5)^2} = \frac{-16x^3}{(2x^4 - 5)^2} \).
6.
FLASHCARD QUESTION
Front
What is the derivative of a constant?
Back
The derivative of a constant is zero. For example, if \( c \) is a constant, then \( \frac{d}{dx}(c) = 0 \).
7.
FLASHCARD QUESTION
Front
Differentiate \( y = \frac{x^2}{3x - 1} \).
Back
Using the Quotient Rule, \( y' = \frac{(2x)(3x - 1) - (x^2)(3)}{(3x - 1)^2} = \frac{3x^2 - 2x}{(3x - 1)^2} \).
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