Differentiability Practice

Take timed practice tests on Differentiability for JEE Main and JEE Advanced with session-wise drills, score review, and explanation-led revision.

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Differentiability Practice Tests

Six 20-question timed sessions plus one 60-question module test. Each question is original and calibrated from the uploaded material pattern without copying PDF wording.

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JEE Main Session 1

3Questions

60s/QTimer

+4 / -1Scoring

Preview all 14 questions in Differentiability Practice Tests (no login required)

Derivative definition
1. The derivative f'(x) is defined as:
- A. lim(h→0) [f(x+h)−f(x)]/h (Correct)
- B. f(x+h)−f(x)
- C. [f(x+1)−f(x)]/1
- D. lim(h→1) [f(x+h)−f(x)]/h
Explanation: First principles definition of the derivative: f'(x) = lim(h→0) [f(x+h)−f(x)]/h, provided this limit exists.
Standard derivatives
2. d/dx (xⁿ) equals:
- A. nxⁿ⁻¹ (Correct)
- B. nxⁿ
- C. xⁿ⁻¹
- D. xⁿ⁺¹/(n+1)
Explanation: Power rule: d/dx(xⁿ) = nxⁿ⁻¹. For example: d/dx(x³) = 3x².
Chain rule
3. d/dx [f(g(x))] equals:
- A. f'(g(x)) × g'(x) (Correct)
- B. f'(x) × g'(x)
- C. f(g'(x))
- D. f'(g(x))
Explanation: Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). Example: d/dx(sin x²) = cos(x²) × 2x.
Product rule
4. d/dx [u × v] equals:
- A. u'v + uv' (Correct)
- B. u'v'
- C. u'v − uv'
- D. u'v/v'
Explanation: Product rule: d/dx(uv) = u'v + uv'. Example: d/dx(x sin x) = sin x + x cos x.
Logarithmic differentiation
5. To differentiate y = xˢⁱⁿ ˣ, we use:
- A. Logarithmic differentiation: take ln, then differentiate (Correct)
- B. Power rule directly
- C. Chain rule only
- D. Product rule directly
Explanation: For y = f(x)^g(x), take ln: ln y = g(x) ln f(x). Differentiate both sides: y'/y = g'(x) ln f(x) + g(x) f'(x)/f(x). Multiply by y for y'.
Implicit differentiation
6. For x² + y² = r², dy/dx equals:
- A. −x/y (Correct)
- B. x/y
- C. −y/x
- D. y/x
Explanation: Implicit differentiation: differentiate both sides w.r.t. x: 2x + 2y(dy/dx) = 0 → dy/dx = −x/y.
Higher order derivatives
7. If y = sin x, then d²y/dx² is:
- A. −sin x (Correct)
- B. cos x
- C. −cos x
- D. sin x
Explanation: y' = cos x. y'' = d/dx(cos x) = −sin x.
Differentiability at a point
8. f(x) = |x| is not differentiable at x = 0 because:
- A. LHD ≠ RHD at x = 0 (−1 ≠ +1) (Correct)
- B. f(0) is undefined
- C. f is discontinuous at 0
- D. f'(0) = ∞
Explanation: Left derivative: lim(h→0⁻) (|h|−0)/h = −1. Right derivative: lim(h→0⁺) h/h = +1. LHD ≠ RHD → not differentiable at x = 0 (though f is continuous there).
JEE Main
9. The derivative of sin⁻¹x is:
- A. 1/√(1−x²) (Correct)
- B. 1/(1+x²)
- C. −1/√(1−x²)
- D. √(1−x²)
Explanation: d/dx(sin⁻¹x) = 1/√(1−x²) for |x|
Parametric differentiation
10. If x = t², y = t³, then dy/dx is:
- A. 3t/2 (Correct)
- B. 3t²/2t = 3t/2
- C. 2t
- D. t²
Explanation: dy/dx = (dy/dt)/(dx/dt) = (3t²)/(2t) = 3t/2.
JEE Advanced
11. If y = tan⁻¹(2x/(1−x²)), then dy/dx for |x| < 1 is:
- A. 2/(1+x²) (Correct)
- B. 1/(1+x²)
- C. 2(1−x²)/(1+x²)²
- D. 1/(1+4x²)
Explanation: Recognize: 2x/(1−x²) = tan(2θ) when x = tan θ. So y = tan⁻¹(tan 2θ) = 2θ = 2tan⁻¹x. dy/dx = 2/(1+x²).
Mixed
12. d/dx(ln x) equals:
- A. 1/x (Correct)
- B. 1
- C. ln x/x
- D. x/ln x
Explanation: d/dx(ln x) = 1/x for x > 0. More generally, d/dx(ln|x|) = 1/x.
Mixed
13. d/dx(eˣ sin x) equals:
- A. eˣ(sin x + cos x)
- B. eˣ sin x + eˣ cos x
- C. eˣ cos x
- D. Both A and B are the same (Correct)
Explanation: Product rule: (eˣ)'sin x + eˣ(sin x)' = eˣ sin x + eˣ cos x = eˣ(sin x + cos x). Options A and B are identical expressions.
Mixed
14. The nth derivative of eˣ is:
- A. eˣ for all n (Correct)
- B. neˣ
- C. eˣ/n!
- D. xⁿeˣ
Explanation: Differentiating eˣ any number of times gives eˣ. Dⁿeˣ = eˣ for all positive integers n.

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JEE Main Session 1

3Questions

60s/QTimer

+4 / -1Scoring

Preview all 14 questions in Differentiability Practice Tests (no login required)

Derivative definition
1. The derivative f'(x) is defined as:
- A. lim(h→0) [f(x+h)−f(x)]/h (Correct)
- B. f(x+h)−f(x)
- C. [f(x+1)−f(x)]/1
- D. lim(h→1) [f(x+h)−f(x)]/h
Explanation: First principles definition of the derivative: f'(x) = lim(h→0) [f(x+h)−f(x)]/h, provided this limit exists.
Standard derivatives
2. d/dx (xⁿ) equals:
- A. nxⁿ⁻¹ (Correct)
- B. nxⁿ
- C. xⁿ⁻¹
- D. xⁿ⁺¹/(n+1)
Explanation: Power rule: d/dx(xⁿ) = nxⁿ⁻¹. For example: d/dx(x³) = 3x².
Chain rule
3. d/dx [f(g(x))] equals:
- A. f'(g(x)) × g'(x) (Correct)
- B. f'(x) × g'(x)
- C. f(g'(x))
- D. f'(g(x))
Explanation: Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). Example: d/dx(sin x²) = cos(x²) × 2x.
Product rule
4. d/dx [u × v] equals:
- A. u'v + uv' (Correct)
- B. u'v'
- C. u'v − uv'
- D. u'v/v'
Explanation: Product rule: d/dx(uv) = u'v + uv'. Example: d/dx(x sin x) = sin x + x cos x.
Logarithmic differentiation
5. To differentiate y = xˢⁱⁿ ˣ, we use:
- A. Logarithmic differentiation: take ln, then differentiate (Correct)
- B. Power rule directly
- C. Chain rule only
- D. Product rule directly
Explanation: For y = f(x)^g(x), take ln: ln y = g(x) ln f(x). Differentiate both sides: y'/y = g'(x) ln f(x) + g(x) f'(x)/f(x). Multiply by y for y'.
Implicit differentiation
6. For x² + y² = r², dy/dx equals:
- A. −x/y (Correct)
- B. x/y
- C. −y/x
- D. y/x
Explanation: Implicit differentiation: differentiate both sides w.r.t. x: 2x + 2y(dy/dx) = 0 → dy/dx = −x/y.
Higher order derivatives
7. If y = sin x, then d²y/dx² is:
- A. −sin x (Correct)
- B. cos x
- C. −cos x
- D. sin x
Explanation: y' = cos x. y'' = d/dx(cos x) = −sin x.
Differentiability at a point
8. f(x) = |x| is not differentiable at x = 0 because:
- A. LHD ≠ RHD at x = 0 (−1 ≠ +1) (Correct)
- B. f(0) is undefined
- C. f is discontinuous at 0
- D. f'(0) = ∞
Explanation: Left derivative: lim(h→0⁻) (|h|−0)/h = −1. Right derivative: lim(h→0⁺) h/h = +1. LHD ≠ RHD → not differentiable at x = 0 (though f is continuous there).
JEE Main
9. The derivative of sin⁻¹x is:
- A. 1/√(1−x²) (Correct)
- B. 1/(1+x²)
- C. −1/√(1−x²)
- D. √(1−x²)
Explanation: d/dx(sin⁻¹x) = 1/√(1−x²) for |x|
Parametric differentiation
10. If x = t², y = t³, then dy/dx is:
- A. 3t/2 (Correct)
- B. 3t²/2t = 3t/2
- C. 2t
- D. t²
Explanation: dy/dx = (dy/dt)/(dx/dt) = (3t²)/(2t) = 3t/2.
JEE Advanced
11. If y = tan⁻¹(2x/(1−x²)), then dy/dx for |x| < 1 is:
- A. 2/(1+x²) (Correct)
- B. 1/(1+x²)
- C. 2(1−x²)/(1+x²)²
- D. 1/(1+4x²)
Explanation: Recognize: 2x/(1−x²) = tan(2θ) when x = tan θ. So y = tan⁻¹(tan 2θ) = 2θ = 2tan⁻¹x. dy/dx = 2/(1+x²).
Mixed
12. d/dx(ln x) equals:
- A. 1/x (Correct)
- B. 1
- C. ln x/x
- D. x/ln x
Explanation: d/dx(ln x) = 1/x for x > 0. More generally, d/dx(ln|x|) = 1/x.
Mixed
13. d/dx(eˣ sin x) equals:
- A. eˣ(sin x + cos x)
- B. eˣ sin x + eˣ cos x
- C. eˣ cos x
- D. Both A and B are the same (Correct)
Explanation: Product rule: (eˣ)'sin x + eˣ(sin x)' = eˣ sin x + eˣ cos x = eˣ(sin x + cos x). Options A and B are identical expressions.
Mixed
14. The nth derivative of eˣ is:
- A. eˣ for all n (Correct)
- B. neˣ
- C. eˣ/n!
- D. xⁿeˣ
Explanation: Differentiating eˣ any number of times gives eˣ. Dⁿeˣ = eˣ for all positive integers n.