Question 1Easy
If y = cos 3x, find d²y/dx².
- A−9 cos 3x
- B9 cos 3x
- C−3 sin 3x
- D−9 sin 3x
Show answer and explanation
Correct answer: (A) −9 cos 3x
Explanation: Step 1: y' = −3 sin 3x. Step 2: y'' = −9 cos 3x.
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 36 questions on this lesson, each with an explanation.
If y = cos 3x, find d²y/dx².
Correct answer: (A) −9 cos 3x
Explanation: Step 1: y' = −3 sin 3x. Step 2: y'' = −9 cos 3x.
If f(x) = x⁴, find f''(x).
Correct answer: (D) 12x²
Explanation: Step 1: f'(x) = 4x³. Step 2: Differentiate again: f''(x) = 12x². ∴ f''(x) = 12x².
If y = 3x⁵ − 2x³ + x, find d³y/dx³.
Correct answer: (A) 180x² − 12
Explanation: Step 1: y' = 15x⁴ − 6x² + 1. Step 2: y'' = 60x³ − 12x. Step 3: y''' = 180x² − 12.
If y = sin x cos x, find the value of d³y/dx³ at x = 0.
Correct answer: (B) −4
Explanation: Step 1: y = ½ sin 2x. Step 2: y' = cos 2x, y'' = −2 sin 2x, y''' = −4 cos 2x. Step 3: At x = 0: y''' = −4.
If y = x cos x, then d²y/dx² + y equals:
Correct answer: (C) −2 sin x
Explanation: Step 1: y' = cos x − x sin x. Step 2: y'' = −sin x − sin x − x cos x = −2 sin x − x cos x. Step 3: y'' + y = −2 sin x − x cos x + x cos x = −2 sin x.
If x = sec θ and y = tan θ, find d²y/dx² at θ = π/4.
Correct answer: (D) −1
Explanation: Step 1: dx/dθ = sec θ tan θ and dy/dθ = sec²θ, so dy/dx = sec θ/tan θ = csc θ. Step 2: d/dθ (csc θ) = −csc θ cot θ. Step 3: d²y/dx² = −csc θ cot θ ÷ (sec θ tan θ) = −cot³θ. Step 4: At θ = π/4: d²y/dx² = −1.
If f(x) = sin x, find f'''(x) (third derivative).
Correct answer: (B) −cos x
Explanation: Step 1: f'(x) = cos x. Step 2: f''(x) = −sin x. Step 3: f'''(x) = −cos x. ∴ f'''(x) = −cos x.
If y = ax³ + bx², where a and b are constants, d³y/dx³ = 12 and the value of d²y/dx² at x = 1 is 10, find b.
Correct answer: (C) −1
Explanation: Step 1: y' = 3ax² + 2bx, y'' = 6ax + 2b, y''' = 6a. Step 2: 6a = 12, so a = 2. Step 3: y''(1) = 6(2)(1) + 2b = 12 + 2b = 10, so b = −1.
If y = cot x, which of the following expresses d²y/dx² in terms of y?
Correct answer: (D) 2y(1 + y²)
Explanation: Step 1: y' = −csc²x = −(1 + cot²x) = −(1 + y²). Step 2: y'' = −2y·y' = −2y·(−(1 + y²)) = 2y(1 + y²); −(1 + y²) is only the first derivative. ∴ 2y(1 + y²)
If x² + y² = 25, find the value of d²y/dx² at the point (3, 4).
Correct answer: (C) −25/64
Explanation: Step 1: 2x + 2y y' = 0 ⇒ y' = −x/y = −3/4 at (3, 4). Step 2: y'' = −(y − x y')/y² = −(x² + y²)/y³ = −25/64 (−3/4 is only the first derivative). ∴ −25/64
The first 10 questions in every subject are free, each with an explanation.