Which rule is used to differentiate f(x) = u(x) / v(x)?
A
Sum Rule
B
Product Rule
C
Chain Rule
D
Quotient Rule
Analysis & Theory
The quotient rule is used to differentiate a function that is the division of two functions.
What is the derivative of f(x) = x / (x + 1)?
A
1 / (x + 1)
B
(x + 1 - x)/(x + 1)^2
C
1 / x
D
(x + 1 + x)/(x + 1)^2
Analysis & Theory
Use quotient rule: [(x+1)·1 − x·1] / (x+1)² = (x+1−x)/(x+1)² = 1/(x+1)².
If f(x) = (sin x) / x, what is f'(x)?
A
cos x / x
B
(x cos x − sin x) / x²
C
(x sin x − cos x) / x²
D
None of the above
Analysis & Theory
Use quotient rule: [x·cos x − sin x] / x².
Differentiate f(x) = (ln x) / x
A
1 / x
B
(x − ln x) / x²
C
(ln x − 1)/x²
D
(1 − ln x)/x²
Analysis & Theory
Use quotient rule: [x·(1/x) − ln x] / x² = (1 − ln x)/x².
What is the derivative of f(x) = x² / e^x?
A
2x / e^x
B
(2x·e^x − x²·e^x) / (e^x)^2
C
(2x − x²) / e^x
D
(2x − x²) / (e^x)^2
Analysis & Theory
Use quotient rule: [e^x·2x − x²·e^x] / (e^x)² = (2x − x²) / e^x.
Find d/dx of (x³) / (x² + 1)
A
(3x²(x² + 1) − x³·2x) / (x² + 1)²
B
(3x² − 2x³) / (x² + 1)²
C
(x³ − x²) / x
D
None of the above
Analysis & Theory
Use quotient rule: [(x² + 1)·3x² − x³·2x] / (x² + 1)².
If f(x) = (e^x) / (ln x), then f'(x) is:
A
(e^x ln x − e^x / x) / (ln x)^2
B
(e^x − ln x) / (ln x)^2
C
(e^x (ln x − 1/x)) / (ln x)^2
D
None of the above
Analysis & Theory
Using quotient rule: [(ln x)·e^x − e^x·(1/x)] / (ln x)² = e^x(ln x − 1/x) / (ln x)².
What is the derivative of f(x) = 1 / x?
A
1
B
−1 / x²
C
ln x
D
x / x²
Analysis & Theory
Derivative of 1/x is −1/x² using basic rule or quotient rule.
Differentiate f(x) = tan x / x
A
(x sec²x − tan x) / x²
B
(sec²x − tan x) / x²
C
sec²x / x
D
None of the above
Analysis & Theory
Quotient rule: [x·sec²x − tan x·1] / x² = (x sec²x − tan x)/x².
If f(x) = x / ln x, then f'(x) is:
A
(ln x − 1)/ (ln x)²
B
(1 − ln x)/(ln x)²
C
(ln x + 1)/(ln x)²
D
None of these
Analysis & Theory
Use quotient rule: [(ln x)(1) − x(1/x)] / (ln x)² = (ln x − 1)/ (ln x)².