Pure Mathematics — Second: Differential & Integral Calculus · Mathematics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 35 questions on this lesson, each with an explanation.
If f(x) = tan(2x), find f'(x).
Correct answer: (A) 2sec²(2x)
Explanation: Step 1: d/dx[tan u] = sec²u·u' with u = 2x. Step 2: u' = 2. Step 3: f'(x) = 2sec²(2x). ∴ f'(x) = 2sec²(2x).
If y = (3x + 1)⁵, find dy/dx.
Correct answer: (C) 15(3x + 1)⁴
Explanation: Step 1: Chain rule: d/dx[u⁵] = 5u⁴·u' with u = 3x + 1. Step 2: u' = 3. Step 3: dy/dx = 5(3x + 1)⁴ · 3 = 15(3x + 1)⁴. ∴ dy/dx = 15(3x + 1)⁴.
If f(x) = x sin x, find f'(x).
Correct answer: (D) sin x + x cos x
Explanation: Step 1: Product rule (uv)' = u'v + uv' with u = x, v = sin x. Step 2: u' = 1, v' = cos x. Step 3: f'(x) = 1·sin x + x·cos x = sin x + x cos x. ∴ f'(x) = sin x + x cos x.
∫ (sec x tan x − 2 sin x) dx = ..........
Correct answer: (B) sec x + 2 cos x + c
Explanation: Step 1: ∫ sec x tan x dx = sec x + c. Step 2: ∫ −2 sin x dx = −2(−cos x) = 2 cos x. Step 3: The integral = sec x + 2 cos x + c.
If f(x) = 3 csc x, then f′(π/6) = ..........
Correct answer: (B) −6√3
Explanation: Step 1: f′(x) = −3 csc x cot x. Step 2: csc π/6 = 2 and cot π/6 = √3. Step 3: f′(π/6) = −3 × 2 × √3 = −6√3.
If y = cot 3x, then dy/dx = ..........
Correct answer: (C) −3 csc²3x
Explanation: Step 1: d/dx (cot u) = −csc²u × du/dx with u = 3x. Step 2: du/dx = 3. Step 3: dy/dx = −3 csc²3x.
If y = sin²x, find dy/dx.
Correct answer: (D) 2sin x cos x
Explanation: Step 1: Write y = (sin x)² and use the chain rule: 2(sin x)·(sin x)'. Step 2: (sin x)' = cos x. Step 3: dy/dx = 2sin x cos x. ∴ dy/dx = 2sin x cos x.
If y = sec²x − tan²x, then dy/dx = ..........
Correct answer: (B) 0
Explanation: Step 1: sec²x − tan²x = 1 for every x in the domain (since 1 + tan²x = sec²x). Step 2: So y = 1, a constant. Step 3: (Check by differentiating: 2 sec x · sec x tan x − 2 tan x · sec²x = 0.) Step 4: dy/dx = 0.
∫ (sin x / cos²x) dx = ..........
Correct answer: (B) sec x + c
Explanation: Step 1: sin x / cos²x = (1/cos x)(sin x / cos x) = sec x tan x. Step 2: ∫ sec x tan x dx = sec x + c.
If y = sin³(2x), find dy/dx.
Correct answer: (A) 6sin²(2x)cos(2x)
Explanation: Step 1: Write y = [sin(2x)]³. Outer chain: 3[sin(2x)]²·(sin(2x))'. Step 2: (sin(2x))' = 2cos(2x). Step 3: dy/dx = 3sin²(2x)·2cos(2x) = 6sin²(2x)cos(2x). ∴ dy/dx = 6sin²(2x)cos(2x).
The first 10 questions in every subject are free, each with an explanation.