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 37 questions on this lesson, each with an explanation.
If y = log x (base 10), find dy/dx.
Correct answer: (C) 1/(x ln 10)
Explanation: Step 1: d/dx[logₐ x] = 1/(x ln a). Step 2: Here a = 10. Step 3: dy/dx = 1/(x ln 10). ∴ dy/dx = 1/(x ln 10).
If y = ln(5x), find dy/dx.
Correct answer: (B) 1/x
Explanation: Step 1: d/dx[ln u] = u'/u with u = 5x. Step 2: u' = 5. Step 3: dy/dx = 5/(5x) = 1/x. ∴ dy/dx = 1/x.
If y = x²e^(3x), find dy/dx.
Correct answer: (A) e^(3x)(3x² + 2x)
Explanation: Step 1: Product rule with u = x², v = e^(3x). Step 2: u' = 2x, v' = 3e^(3x) (chain rule). Step 3: dy/dx = 2x·e^(3x) + x²·3e^(3x). Step 4: Factor e^(3x): dy/dx = e^(3x)(3x² + 2x). ∴ dy/dx = e^(3x)(3x² + 2x).
If y = e⁻ˣ cos x, find d²y/dx².
Correct answer: (B) 2e⁻ˣ sin x
Explanation: Step 1: y' = −e⁻ˣ cos x − e⁻ˣ sin x = −e⁻ˣ(cos x + sin x). Step 2: y'' = e⁻ˣ(cos x + sin x) − e⁻ˣ(−sin x + cos x). Step 3: y'' = 2e⁻ˣ sin x.
If y = (cos x)ˣ where 0 < x < π/2, then dy/dx = ..........
Correct answer: (D) (cos x)ˣ (ln cos x − x tan x)
Explanation: Step 1: ln y = x ln cos x. Step 2: (1/y) · dy/dx = 1 × ln cos x + x × (−sin x/cos x) = ln cos x − x tan x. Step 3: Multiply both sides by y = (cos x)ˣ. ∴ dy/dx = (cos x)ˣ (ln cos x − x tan x). The power rule x(cos x)ˣ⁻¹(−sin x) is wrong because the exponent is not constant.
If y = ln(sin x), find dy/dx.
Correct answer: (D) cot x
Explanation: Step 1: d/dx[ln u] = u'/u with u = sin x. Step 2: u' = cos x. Step 3: dy/dx = cos x/sin x = cot x. ∴ dy/dx = cot x.
If y = x^(sin x) where x > 0, then the value of dy/dx at x = π/2 equals ..........
Correct answer: (A) 1
Explanation: Step 1: ln y = sin x · ln x ⇒ (1/y) · dy/dx = cos x · ln x + (sin x)/x. Step 2: At x = π/2: y = (π/2)¹ = π/2, cos(π/2) = 0, sin(π/2) = 1. Step 3: dy/dx = (π/2) × (0 + 1/(π/2)) = (π/2) × (2/π) = 1. ∴ 1 (2/π is the value of (1/y) · dy/dx before multiplying by y).
If y = e^(cos 2x), find dy/dx.
Correct answer: (D) −2sin(2x)e^(cos 2x)
Explanation: Step 1: d/dx[eu] = eu·u' with u = cos 2x. Step 2: u' = −sin(2x)·2 = −2sin(2x). Step 3: dy/dx = −2sin(2x)e^(cos 2x). ∴ dy/dx = −2sin(2x)e^(cos 2x).
If y = xx (x greater than 0), find dy/dx.
Correct answer: (B) xx(ln x + 1)
Explanation: Step 1: Take ln: ln y = x ln x. Step 2: Differentiate: (1/y)·y' = ln x + x·(1/x) = ln x + 1. Step 3: y' = y(ln x + 1) = xx(ln x + 1). ∴ dy/dx = xx(ln x + 1).
If y = ln(sec x + tan x), find the value of d²y/dx² at x = π/4.
Correct answer: (C) √2
Explanation: Step 1: dy/dx = (sec x tan x + sec²x)/(sec x + tan x) = sec x(tan x + sec x)/(sec x + tan x) = sec x. Step 2: d²y/dx² = sec x tan x. ∴ At x = π/4: √2 × 1 = √2; the value 2 is sec²(π/4), which comes from differentiating tan x instead of sec x.
The first 10 questions in every subject are free, each with an explanation.