limₓ→₀ (s∈ x)/x = 1
limₓ→₀ (1 - cos x)/x = 0
limₓ→₀ (tan x)/x = 1
d/dx (xⁿ) = nxⁿ⁻¹
d/dx (s∈ x) = cos x
d/dx (cos x) = -s∈ x
d/dx (tan x) = sec²x
d/dx (eˣ) = eˣ
d/dx (logₐx) = 1/(x ln a)
(dy/dx) = (dy/du) × (du/dx)
d²y/dx² = d/dx (dy/dx)
f(x) = xⁿ, f'(x) = nxⁿ⁻¹
f(x) = s∈ x, f'(x) = cos x
f(x) = cos x, f'(x) = -s∈ x
f(x) = eˣ, f'(x) = eˣ
f(x) = logₐx, f'(x) = 1/(x ln a)

🪤 The 5 Mistakes That Cost Marks

Not using the definition of a derivative to find f'(x)
Forgetting to apply the cha∈ rule when differentiating composite functions
Not using the product rule and quotient rule when differentiating products and quotients
Forgetting to check for continuity and differentiability at a point
Not applying L'Hopital's rule when evaluating indeterminate forms

✏️ 3 Solved PYQs

Find the derivative of f(x) = x³ s∈ x Step 1: Apply the product rule, f'(x) = (x³)' s∈ x + x³ (s∈ x)' Step 2: Evaluate the derivatives, f'(x) = 3x² s∈ x + x³ cos x
Evaluate the limit limₓ→₀ (eˣ - 1)/x Step 1: Apply L'Hopital's rule, limₓ→₀ (eˣ - 1)/x = limₓ→₀ (eˣ)' / (x)' Step 2: Evaluate the derivatives, limₓ→₀ (eˣ - 1)/x = limₓ→₀ eˣ / 1 = 1
Find the derivative of f(x) = logₐx Step 1: Apply the definition of a derivative, f'(x) = limₕ→₀ (logₐ(x + h) - logₐx) / h Step 2: Simplify the expression, f'(x) = limₕ→₀ (logₐ(1 + h/x)) / h Step 3: Apply L'Hopital's rule, f'(x) = limₕ→₀ (logₐ(1 + h/x))' / h' Step 4: Evaluate the derivatives, f'(x) = 1/(x ln a)

🧠 The One Thing Most Students Get Wrong

Most students get wrong the application of L'Hopital's rule to evaluate indeterminate forms
- L'Hopital's rule can only be applied when the limit is ∈ the form 0/0 or ∞/∞
- The rule states that limₓ→a (f(x)/g(x)) = limₓ→a (f'(x)/g'(x))

👁️ Ayush's Note

To solve limits and derivatives problems, first identify the type of problem
- If it's a limit problem, check if it's ∈ the form 0/0 or ∞/∞ and apply L'Hopital's rule if necessary
- Is it's a derivative problem, apply the definition of a derivative or use the product rule, quotient rule, or cha∈ rule as needed
Practice, practice, practice, as the more you practice, the more comfortable you'll become with the formulas and techniques

🔁 Last 5 Minutes Box

Make sure to check your work and review your answers
Use the last 5 minutes to go through the exam compass and make sure you've answered all the questions
Don't leave any question blank, as you'll get marks for attempting it
Use the JEE Advanced and NEET level shortcuts to save time and increase your score

📝 Practice MCQs

1. What is the derivative of f(x) = x² s∈ x?

A) 2x s∈ x + x² cos x

B) 2x cos x - x² s∈ x

C) x² cos x - 2x s∈ x

D) 2x s∈ x - x² cos x

Answer: A) 2x s∈ x + x² cos x.

2. Evaluate the limit limₓ→₀ (1 - cos x)/x

A) 0

B) 1

C) ∞

D) -1

Answer: A) 0.

3. What is the derivative of f(x) = eˣ?

A) eˣ

B) -eˣ

C) 2eˣ

D) 1/eˣ

Answer: A) eˣ.

4. Evaluate the limit limₓ→∞ (1 + 1/x)ˣ

A) 0

B) 1

C) e

D) ∞

Answer: C) e.

5. What is the derivative of f(x) = logₐx?

A) 1/(x ln a)

B) -1/(x ln a)

C) 1/(x log a)

D) -1/(x log a)

Answer: A) 1/(x ln a).

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

📚 Academic References

Content verified against peer-reviewed research:

Phasor measurement units, WAMS, and their applications in protect... — Journal of Modern Power Systems and Clean Energy (2018) 🔓 — DOI ↗
The Opaque Nature of Intelligence and the Pursuit of Explainable ... — Academic Journal (2023) 🔓 — DOI ↗
The Primacy of Phenomenology Over Cognitivism. Towards a Critique... — Online Publication Service of Würzburg University (Würzburg University) (2015) 🔓 — DOI ↗

🔓 = Open Access article

🎬 Watch video explanations on YouTube →

This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides:

limₓ→₀ (s∈ x)/x = 1
limₓ→₀ (1 - cos x)/x = 0
limₓ→₀ (tan x)/x = 1
d/dx (xⁿ) = nxⁿ⁻¹
d/dx (s∈ x) = cos x
d/dx (cos x) = -s∈ x
d/dx (tan x) = sec²x
d/dx (eˣ) = eˣ
d/dx (logₐx) = 1/(x ln a)
(dy/dx) = (dy/du) × (du/dx)
d²y/dx² = d/dx (dy/dx)
f(x) = xⁿ, f'(x) = nxⁿ⁻¹
f(x) = s∈ x, f'(x) = cos x
f(x) = cos x, f'(x) = -s∈ x
f(x) = eˣ, f'(x) = eˣ
f(x) = logₐx, f'(x) = 1/(x ln a)

🪤 The 5 Mistakes That Cost Marks

Not using the definition of a derivative to find f'(x)
Forgetting to apply the cha∈ rule when differentiating composite functions
Not using the product rule and quotient rule when differentiating products and quotients
Forgetting to check for continuity and differentiability at a point
Not applying L'Hopital's rule when evaluating indeterminate forms

✏️ 3 Solved PYQs

Find the derivative of f(x) = x³ s∈ x Step 1: Apply the product rule, f'(x) = (x³)' s∈ x + x³ (s∈ x)' Step 2: Evaluate the derivatives, f'(x) = 3x² s∈ x + x³ cos x
Evaluate the limit limₓ→₀ (eˣ - 1)/x Step 1: Apply L'Hopital's rule, limₓ→₀ (eˣ - 1)/x = limₓ→₀ (eˣ)' / (x)' Step 2: Evaluate the derivatives, limₓ→₀ (eˣ - 1)/x = limₓ→₀ eˣ / 1 = 1
Find the derivative of f(x) = logₐx Step 1: Apply the definition of a derivative, f'(x) = limₕ→₀ (logₐ(x + h) - logₐx) / h Step 2: Simplify the expression, f'(x) = limₕ→₀ (logₐ(1 + h/x)) / h Step 3: Apply L'Hopital's rule, f'(x) = limₕ→₀ (logₐ(1 + h/x))' / h' Step 4: Evaluate the derivatives, f'(x) = 1/(x ln a)

🧠 The One Thing Most Students Get Wrong

Most students get wrong the application of L'Hopital's rule to evaluate indeterminate forms
- L'Hopital's rule can only be applied when the limit is ∈ the form 0/0 or ∞/∞
- The rule states that limₓ→a (f(x)/g(x)) = limₓ→a (f'(x)/g'(x))

👁️ Ayush's Note

To solve limits and derivatives problems, first identify the type of problem
- If it's a limit problem, check if it's ∈ the form 0/0 or ∞/∞ and apply L'Hopital's rule if necessary
- Is it's a derivative problem, apply the definition of a derivative or use the product rule, quotient rule, or cha∈ rule as needed
Practice, practice, practice, as the more you practice, the more comfortable you'll become with the formulas and techniques

🔁 Last 5 Minutes Box

Make sure to check your work and review your answers
Use the last 5 minutes to go through the exam compass and make sure you've answered all the questions
Don't leave any question blank, as you'll get marks for attempting it
Use the JEE Advanced and NEET level shortcuts to save time and increase your score

📝 Practice MCQs

1. What is the derivative of f(x) = x² s∈ x?

A) 2x s∈ x + x² cos x

B) 2x cos x - x² s∈ x

C) x² cos x - 2x s∈ x

D) 2x s∈ x - x² cos x

Answer: A) 2x s∈ x + x² cos x.

2. Evaluate the limit limₓ→₀ (1 - cos x)/x

A) 0

B) 1

C) ∞

D) -1

Answer: A) 0.

3. What is the derivative of f(x) = eˣ?

A) eˣ

B) -eˣ

C) 2eˣ

D) 1/eˣ

Answer: A) eˣ.

4. Evaluate the limit limₓ→∞ (1 + 1/x)ˣ

A) 0

B) 1

C) e

D) ∞

Answer: C) e.

5. What is the derivative of f(x) = logₐx?

A) 1/(x ln a)

B) -1/(x ln a)

C) 1/(x log a)

D) -1/(x log a)

Answer: A) 1/(x ln a).

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

📚 Academic References

Content verified against peer-reviewed research:

Phasor measurement units, WAMS, and their applications in protect... — Journal of Modern Power Systems and Clean Energy (2018) 🔓 — DOI ↗
The Opaque Nature of Intelligence and the Pursuit of Explainable ... — Academic Journal (2023) 🔓 — DOI ↗
The Primacy of Phenomenology Over Cognitivism. Towards a Critique... — Online Publication Service of Würzburg University (Würzburg University) (2015) 🔓 — DOI ↗

🔓 = Open Access article

🎬 Watch video explanations on YouTube →

This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides: