Question 1Easy
If xy = 4, find dy/dx.
- Ay/x
- B−x/y
- C−1
- D−y/x
Show answer and explanation
Correct answer: (D) −y/x
Explanation: Step 1: Differentiate both sides with respect to x: x dy/dx + y = 0. Step 2: dy/dx = −y/x.
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 34 questions on this lesson, each with an explanation.
If xy = 4, find dy/dx.
Correct answer: (D) −y/x
Explanation: Step 1: Differentiate both sides with respect to x: x dy/dx + y = 0. Step 2: dy/dx = −y/x.
If x = cos³θ and y = sin³θ, find dy/dx at θ = π/6.
Correct answer: (B) −1/√3
Explanation: Step 1: dx/dθ = −3cos²θ sin θ and dy/dθ = 3sin²θ cos θ. Step 2: dy/dx = 3sin²θ cos θ ÷ (−3cos²θ sin θ) = −tan θ. Step 3: At θ = π/6: dy/dx = −1/√3.
A curve is given by x = t², y = t³. Find dy/dx.
Correct answer: (A) 3t/2
Explanation: Step 1: dx/dt = 2t and dy/dt = 3t². Step 2: dy/dx = (dy/dt)/(dx/dt) = 3t²/(2t). Step 3: Simplify: dy/dx = 3t/2. ∴ dy/dx = 3t/2.
The point A lies on the curve xy + y² = 2x, its x-coordinate is 1 and its y-coordinate is positive. Find the slope of the tangent to the curve at A.
Correct answer: (B) 1/3
Explanation: Step 1: Put x = 1: y + y² = 2, so (y + 2)(y − 1) = 0 and y = 1 (positive). A = (1, 1). Step 2: Differentiating implicitly: y + x dy/dx + 2y dy/dx = 2, so dy/dx = (2 − y)/(x + 2y). Step 3: At A: dy/dx = 1/3.
Find the points of the curve x = 2t³ − 3t² − 12t, y = t² + 1 at which the tangent is vertical.
Correct answer: (C) (7, 2) and (−20, 5)
Explanation: Step 1: The tangent is vertical when dx/dt = 0 and dy/dt ≠ 0. Step 2: dx/dt = 6t² − 6t − 12 = 6(t − 2)(t + 1) = 0, so t = 2 or t = −1 (dy/dt = 2t ≠ 0 there). Step 3: t = 2: x = 16 − 12 − 24 = −20, y = 5. t = −1: x = −2 − 3 + 12 = 7, y = 2. Step 4: The points are (7, 2) and (−20, 5).
Find the derivative of (x³ + 1) with respect to (x² − 1) at x = 2.
Correct answer: (C) 3
Explanation: Step 1: Put y = x³ + 1 and z = x² − 1, so dy/dx = 3x² and dz/dx = 2x. Step 2: dy/dz = 3x²/(2x) = 3x/2. Step 3: At x = 2: dy/dz = 3.
If x² + y² = 25, find dy/dx by implicit differentiation.
Correct answer: (A) −x/y
Explanation: Step 1: Differentiate both sides w.r.t. x: 2x + 2y·(dy/dx) = 0. Step 2: 2y·(dy/dx) = −2x. Step 3: dy/dx = −2x/(2y) = −x/y. ∴ dy/dx = −x/y.
For the curve x = t² − 4t, y = t³ − 12t, find the value(s) of t at which the curve has a horizontal tangent.
Correct answer: (A) t = −2 only
Explanation: Step 1: dx/dt = 2t − 4 and dy/dt = 3t² − 12 = 3(t − 2)(t + 2). Step 2: dy/dt = 0 at t = ±2. At t = −2, dx/dt = −8 ≠ 0, so dy/dx = 0: horizontal tangent. Step 3: At t = 2 also dx/dt = 0, so we cannot use dy/dt = 0 there: dy/dx = 3(t + 2)/2 → 6 as t → 2, not zero. Step 4: So the tangent is horizontal at t = −2 only.
If sin(xy) = x, find dy/dx.
Correct answer: (C) (1 − y cos(xy))/(x cos(xy))
Explanation: Step 1: Differentiate: cos(xy)·(y + x·y') = 1 (chain + product on xy). Step 2: y cos(xy) + x cos(xy)·y' = 1. Step 3: y' = (1 − y cos(xy))/(x cos(xy)). ∴ dy/dx = (1 − y cos(xy))/(x cos(xy)).
A cycloid is given by x = t − sin t, y = 1 − cos t. Find dy/dx.
Correct answer: (B) sin t/(1 − cos t)
Explanation: Step 1: dx/dt = 1 − cos t and dy/dt = sin t. Step 2: dy/dx = (dy/dt)/(dx/dt). Step 3: dy/dx = sin t/(1 − cos t). ∴ dy/dx = sin t/(1 − cos t).
The first 10 questions in every subject are free, each with an explanation.