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)
d/dx (aˣ) = aˣ ln a
∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C
∫s∈ x dx = -cos x + C
∫cos x dx = s∈ x + C
∫eˣ dx = eˣ + C
∫(1/x) dx = log|x| + C
(dy/dx) = (dy/du) × (du/dx)
(dy/dx) = (1/(dx/du)) × (dy/du)

🪤 The 5 Mistakes That Cost Marks

Forgetting to apply the cha∈ rule ∈ differentiation
Not using the correct formula for the derivative of trigonometric functions
Failing to simplify the expression before evaluating the limit
Not using L'Hospital's rule when the limit is ∈ the form 0/0 or ∞/∞
Incorrectly applying the product rule or quotient rule ∈ differentiation

✏️ 3 Solved PYQs

Find the derivative of x³ s∈ x
- Using the product rule, (dy/dx) = d/dx (x³) × s∈ x + x³ × d/dx (s∈ x)
- (dy/dx) = 3x² s∈ x + x³ cos x
Evaluate the limit of (s∈ x)/x as x approaches 0
- Using L'Hospital's rule, limₓ→₀ (s∈ x)/x = limₓ→₀ (cos x)/1 = 1
Find the integral of x² s∈ x dx
- Using integration by parts, ∫x² s∈ x dx = -x² cos x + ∫2x cos x dx
- ∫x² s∈ x dx = -x² cos x + 2x s∈ x + 2 ∫s∈ x dx
- ∫x² s∈ x dx = -x² cos x + 2x s∈ x - 2 cos x + C

🧠 The One Thing Most Students Get Wrong

Most students struggle with applying the cha∈ rule and product rule correctly ∈ differentiation
They often forget to multiply by the derivative of the inner function when applying the cha∈ rule
For example, ∈ the derivative of s∈ (2x), they might forget to multiply by the derivative of 2x, which is 2

👁️ Ayush's Note

To master limits and derivatives, practice is key
Start with simple problems and gradually move on to more complex ones
Use online resources and video lectures to supplement your learning
Make sure to review and practice regularly to reinforce your understanding
Focus on developing problem-solving skills and learning to apply formulas and theorems to different types of problems

🔁 Last 5 Minutes Box

Quickly review the formula bank and make sure you can recall all the important formulas
Go through the 5 mistakes that cost marks and make a mental note to avoid them
Take a few deep breaths and try to relax - a clear mind is essential for performing well ∈ the exam
Visualize yourself acing the exam and feeling confident and prepared
Take a final glance at your notes and make sure you have all the necessary materials before heading into the exam

📝 Practice MCQs

1. What is the derivative of x⁴?

A) 2x³

B) 3x²

C) 4x³

D) 5x⁴

Answer: C) 4x³. Explanation: Using the power rule of differentiation, d/dx (xⁿ) = nxⁿ⁻¹.

2. Evaluate the limit of (1 - cos x)/x as x approaches 0

A) 0

B) 1

C) ∞

D) -1

Answer: A) 0. Explanation: Using L'Hospital's rule, limₓ→₀ (1 - cos x)/x = limₓ→₀ (s∈ x)/1 = 0

3. Find the integral of eˣ dx

A) eˣ + C

B) eˣ - C

C) e⁻ˣ + C

D) e⁻ˣ - C

Answer: A) eˣ + C. Explanation: The integral of eˣ is eˣ + C, where C is the constant of integration

4. What is the derivative of s∈ (2x)?

A) 2 cos x

B) 2 cos (2x)

C) cos (2x)

D) -2 s∈ (2x)

Answer: B) 2 cos (2x). Explanation: Using the cha∈ rule, d/dx (s∈ (2x)) = cos (2x) × d/dx (2x) = 2 cos (2x)

5. Evaluate the limit of (tan x)/x as x approaches 0

A) 0

B) 1

C) ∞

D) -1

Answer: B) 1. Explanation: Using L'Hospital's rule, limₓ→₀ (tan x)/x = limₓ→₀ (sec²x)/1 = 1

🚀 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)
d/dx (aˣ) = aˣ ln a
∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C
∫s∈ x dx = -cos x + C
∫cos x dx = s∈ x + C
∫eˣ dx = eˣ + C
∫(1/x) dx = log|x| + C
(dy/dx) = (dy/du) × (du/dx)
(dy/dx) = (1/(dx/du)) × (dy/du)

🪤 The 5 Mistakes That Cost Marks

Forgetting to apply the cha∈ rule ∈ differentiation
Not using the correct formula for the derivative of trigonometric functions
Failing to simplify the expression before evaluating the limit
Not using L'Hospital's rule when the limit is ∈ the form 0/0 or ∞/∞
Incorrectly applying the product rule or quotient rule ∈ differentiation

✏️ 3 Solved PYQs

Find the derivative of x³ s∈ x
- Using the product rule, (dy/dx) = d/dx (x³) × s∈ x + x³ × d/dx (s∈ x)
- (dy/dx) = 3x² s∈ x + x³ cos x
Evaluate the limit of (s∈ x)/x as x approaches 0
- Using L'Hospital's rule, limₓ→₀ (s∈ x)/x = limₓ→₀ (cos x)/1 = 1
Find the integral of x² s∈ x dx
- Using integration by parts, ∫x² s∈ x dx = -x² cos x + ∫2x cos x dx
- ∫x² s∈ x dx = -x² cos x + 2x s∈ x + 2 ∫s∈ x dx
- ∫x² s∈ x dx = -x² cos x + 2x s∈ x - 2 cos x + C

🧠 The One Thing Most Students Get Wrong

Most students struggle with applying the cha∈ rule and product rule correctly ∈ differentiation
They often forget to multiply by the derivative of the inner function when applying the cha∈ rule
For example, ∈ the derivative of s∈ (2x), they might forget to multiply by the derivative of 2x, which is 2

👁️ Ayush's Note

To master limits and derivatives, practice is key
Start with simple problems and gradually move on to more complex ones
Use online resources and video lectures to supplement your learning
Make sure to review and practice regularly to reinforce your understanding
Focus on developing problem-solving skills and learning to apply formulas and theorems to different types of problems

🔁 Last 5 Minutes Box

Quickly review the formula bank and make sure you can recall all the important formulas
Go through the 5 mistakes that cost marks and make a mental note to avoid them
Take a few deep breaths and try to relax - a clear mind is essential for performing well ∈ the exam
Visualize yourself acing the exam and feeling confident and prepared
Take a final glance at your notes and make sure you have all the necessary materials before heading into the exam

📝 Practice MCQs

1. What is the derivative of x⁴?

A) 2x³

B) 3x²

C) 4x³

D) 5x⁴

Answer: C) 4x³. Explanation: Using the power rule of differentiation, d/dx (xⁿ) = nxⁿ⁻¹.

2. Evaluate the limit of (1 - cos x)/x as x approaches 0

A) 0

B) 1

C) ∞

D) -1

Answer: A) 0. Explanation: Using L'Hospital's rule, limₓ→₀ (1 - cos x)/x = limₓ→₀ (s∈ x)/1 = 0

3. Find the integral of eˣ dx

A) eˣ + C

B) eˣ - C

C) e⁻ˣ + C

D) e⁻ˣ - C

Answer: A) eˣ + C. Explanation: The integral of eˣ is eˣ + C, where C is the constant of integration

4. What is the derivative of s∈ (2x)?

A) 2 cos x

B) 2 cos (2x)

C) cos (2x)

D) -2 s∈ (2x)

Answer: B) 2 cos (2x). Explanation: Using the cha∈ rule, d/dx (s∈ (2x)) = cos (2x) × d/dx (2x) = 2 cos (2x)

5. Evaluate the limit of (tan x)/x as x approaches 0

A) 0

B) 1

C) ∞

D) -1

Answer: B) 1. Explanation: Using L'Hospital's rule, limₓ→₀ (tan x)/x = limₓ→₀ (sec²x)/1 = 1

🚀 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: