MATH 141 — CALCULUS I — EXAM 2 2026
with correct answers in bold and
rationale graded A+ new!!
SECTION A: DERIVATIVE RULES (Questions 1–25)
1. Find the derivative of f(x) = 3x⁴ − 5x² + 7x − 2.
A) 12x³ − 10x + 7
B) 12x³ − 5x + 7
C) 12x³ + 10x + 7
D) 3x³ − 10x + 7
Answer: A
Rationale: Apply the power rule term by term: d/dx(3x⁴) = 12x³, d/dx(−5x²) = −10x, d/dx(7x) = 7,
d/dx(−2) = 0.
2. Find f'(x) if f(x) = x⁵ · sin(x).
A) 5x⁴ sin(x) + x⁵ cos(x)
B) 5x⁴ cos(x)
C) x⁵ cos(x) + 5x⁴ sin(x)
D) 5x⁴ sin(x) − x⁵ cos(x)
Answer: A (same as C — both are correct forms)
Rationale: Product rule: (uv)' = u'v + uv'. u = x⁵ → u' = 5x⁴; v = sin x → v' = cos x. So f' = 5x⁴ sin x + x⁵
cos x.
3. Find the derivative of f(x) = eˣ · ln(x).
,A) eˣ(ln x + 1/x)
B) eˣ ln x + 1/x
C) eˣ/x
D) eˣ(1 + ln x)
Answer: A
Rationale: Product rule: u = eˣ, v = ln x. u' = eˣ, v' = 1/x. f' = eˣ ln x + eˣ(1/x) = eˣ(ln x + 1/x).
4. Find f'(x) if f(x) = (x² + 1)/(x − 3).
A) (x² − 6x − 1)/(x − 3)²
B) (2x(x − 3) − (x² + 1))/(x − 3)²
C) (x² + 6x − 1)/(x − 3)²
D) (x² − 6x + 1)/(x − 3)²
Answer: A
Rationale: Quotient rule: (u/v)' = (u'v − uv')/v². u = x²+1, u'=2x; v=x−3, v'=1. Numerator: 2x(x−3) −
(x²+1)(1) = 2x²−6x−x²−1 = x²−6x−1.
5. Find the derivative of f(x) = tan(x).
A) sec²(x)
B) −csc²(x)
C) cot(x)
D) sec(x)tan(x)
Answer: A
Rationale: Standard derivative: d/dx[tan x] = sec²x. (Can also derive via quotient rule: sin x / cos x.)
6. Find f'(x) if f(x) = √(x² + 4).
,A) x/√(x² + 4)
B) 1/(2√(x² + 4))
C) 2x/√(x² + 4)
D) x/(x² + 4)
Answer: A
Rationale: Chain rule: f = (x²+4)^(1/2). f' = (1/2)(x²+4)^(−1/2) · 2x = x/√(x²+4).
7. Find the derivative of f(x) = sin(3x² + 1).
A) cos(3x² + 1)
B) 6x cos(3x² + 1)
C) cos(3x² + 1) · 6x
D) Both B and C
Answer: D
Rationale: Chain rule: outer = sin(u), inner u = 3x²+1. f' = cos(u) · u' = cos(3x²+1) · 6x.
8. Find f'(x) if f(x) = e^(−x²).
A) −2xe^(−x²)
B) e^(−2x)
C) −xe^(−x²)
D) 2xe^(−x²)
Answer: A
Rationale: Chain rule: derivative of e^u is e^u · u'. u = −x², u' = −2x. So f' = e^(−x²)(−2x) = −2xe^(−x²).
9. Find the derivative of f(x) = ln(x³ + 2x).
, A) 1/(x³ + 2x)
B) (3x² + 2)/(x³ + 2x)
C) 3x² + 2
D) ln(3x² + 2)
Answer: B
Rationale: Chain rule for ln: d/dx[ln(u)] = u'/u. u = x³+2x, u' = 3x²+2. So f' = (3x²+2)/(x³+2x).
10. Find f'(x) if f(x) = (sin x)³.
A) 3 sin²x cos x
B) 3 sin²x
C) cos(3 sin²x)
D) 3 cos²x sin x
Answer: A
Rationale: Chain rule: outer = u³, inner u = sin x. f' = 3u² · u' = 3 sin²x · cos x.
11. Find the derivative of f(x) = x · e^(2x).
A) e^(2x)(1 + 2x)
B) e^(2x) + 2x
C) 2xe^(2x)
D) e^(2x)(2x)
Answer: A
Rationale: Product rule: u = x, v = e^(2x). u' = 1, v' = 2e^(2x). f' = 1·e^(2x) + x·2e^(2x) = e^(2x)(1 + 2x).
12. Find f'(x) if f(x) = cos²(x).
A) −2 cos x sin x
with correct answers in bold and
rationale graded A+ new!!
SECTION A: DERIVATIVE RULES (Questions 1–25)
1. Find the derivative of f(x) = 3x⁴ − 5x² + 7x − 2.
A) 12x³ − 10x + 7
B) 12x³ − 5x + 7
C) 12x³ + 10x + 7
D) 3x³ − 10x + 7
Answer: A
Rationale: Apply the power rule term by term: d/dx(3x⁴) = 12x³, d/dx(−5x²) = −10x, d/dx(7x) = 7,
d/dx(−2) = 0.
2. Find f'(x) if f(x) = x⁵ · sin(x).
A) 5x⁴ sin(x) + x⁵ cos(x)
B) 5x⁴ cos(x)
C) x⁵ cos(x) + 5x⁴ sin(x)
D) 5x⁴ sin(x) − x⁵ cos(x)
Answer: A (same as C — both are correct forms)
Rationale: Product rule: (uv)' = u'v + uv'. u = x⁵ → u' = 5x⁴; v = sin x → v' = cos x. So f' = 5x⁴ sin x + x⁵
cos x.
3. Find the derivative of f(x) = eˣ · ln(x).
,A) eˣ(ln x + 1/x)
B) eˣ ln x + 1/x
C) eˣ/x
D) eˣ(1 + ln x)
Answer: A
Rationale: Product rule: u = eˣ, v = ln x. u' = eˣ, v' = 1/x. f' = eˣ ln x + eˣ(1/x) = eˣ(ln x + 1/x).
4. Find f'(x) if f(x) = (x² + 1)/(x − 3).
A) (x² − 6x − 1)/(x − 3)²
B) (2x(x − 3) − (x² + 1))/(x − 3)²
C) (x² + 6x − 1)/(x − 3)²
D) (x² − 6x + 1)/(x − 3)²
Answer: A
Rationale: Quotient rule: (u/v)' = (u'v − uv')/v². u = x²+1, u'=2x; v=x−3, v'=1. Numerator: 2x(x−3) −
(x²+1)(1) = 2x²−6x−x²−1 = x²−6x−1.
5. Find the derivative of f(x) = tan(x).
A) sec²(x)
B) −csc²(x)
C) cot(x)
D) sec(x)tan(x)
Answer: A
Rationale: Standard derivative: d/dx[tan x] = sec²x. (Can also derive via quotient rule: sin x / cos x.)
6. Find f'(x) if f(x) = √(x² + 4).
,A) x/√(x² + 4)
B) 1/(2√(x² + 4))
C) 2x/√(x² + 4)
D) x/(x² + 4)
Answer: A
Rationale: Chain rule: f = (x²+4)^(1/2). f' = (1/2)(x²+4)^(−1/2) · 2x = x/√(x²+4).
7. Find the derivative of f(x) = sin(3x² + 1).
A) cos(3x² + 1)
B) 6x cos(3x² + 1)
C) cos(3x² + 1) · 6x
D) Both B and C
Answer: D
Rationale: Chain rule: outer = sin(u), inner u = 3x²+1. f' = cos(u) · u' = cos(3x²+1) · 6x.
8. Find f'(x) if f(x) = e^(−x²).
A) −2xe^(−x²)
B) e^(−2x)
C) −xe^(−x²)
D) 2xe^(−x²)
Answer: A
Rationale: Chain rule: derivative of e^u is e^u · u'. u = −x², u' = −2x. So f' = e^(−x²)(−2x) = −2xe^(−x²).
9. Find the derivative of f(x) = ln(x³ + 2x).
, A) 1/(x³ + 2x)
B) (3x² + 2)/(x³ + 2x)
C) 3x² + 2
D) ln(3x² + 2)
Answer: B
Rationale: Chain rule for ln: d/dx[ln(u)] = u'/u. u = x³+2x, u' = 3x²+2. So f' = (3x²+2)/(x³+2x).
10. Find f'(x) if f(x) = (sin x)³.
A) 3 sin²x cos x
B) 3 sin²x
C) cos(3 sin²x)
D) 3 cos²x sin x
Answer: A
Rationale: Chain rule: outer = u³, inner u = sin x. f' = 3u² · u' = 3 sin²x · cos x.
11. Find the derivative of f(x) = x · e^(2x).
A) e^(2x)(1 + 2x)
B) e^(2x) + 2x
C) 2xe^(2x)
D) e^(2x)(2x)
Answer: A
Rationale: Product rule: u = x, v = e^(2x). u' = 1, v' = 2e^(2x). f' = 1·e^(2x) + x·2e^(2x) = e^(2x)(1 + 2x).
12. Find f'(x) if f(x) = cos²(x).
A) −2 cos x sin x
