Continuity at x = a: limₓ→an f(x) = f(a)
Differentiability at x = a: f'(a) = limₕ→₀ (f(a + h) - f(a))/h
If f(x) is differentiable at x = a then f(x) is continuous at x = a bit the converse is not always true
If f(x) and g(x) are continuous at x = a then f(x) + g(x), f(x) - g(x), f(x) × g(x), and f(x)/g(x) are continuous at x = an
If f(x) and g(x) are differentiable at x = a then f(x) + g(x), f(x) - g(x), f(x) × g(x), and f(x)/g(x) are differentiable at x = a
Cha∈ Rule: (f(g(x)))' = f'(g(x)) × g'(x)
Rolle's Theorem: if f(x) is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b), then there ∃ c ∈ (a, b) such that f'(c) = 0
Mean Value Theorem: if f(x) is continuous on [a, b] and differentiable on (a, b), then there ∃ c ∈ (a, b) such that f'(c) = (f(b) - f(a))/(b - a)

🪤 The 5 Mistakes That Cost Marks

Not checking the continuity of the function before differentiating
Forgetting to apply the cha∈ rule when differentiating composite functions
Not using the correct formula for the derivative of a function
Not considering the doma∈ of the function when finding the derivative
Not using Rolle's Theorem and Mean Value Theorem to solve problems

✏️ 3 Solved PYQs

Find the derivative of f(x) = (x² + 1)/(x + 1) Step 1: Apply the quotient rule Step 2: Simplify the expression Answer: f'(x) = (2x(x + 1) - (x² + 1))/(x + 1)²
If f(x) is continuous on [0, 1] and differentiable on (0, 1), and f(0) = f(1), then find the value of c ∈ (0, 1) such that f'(c) = 0 Step 1: Apply Rolle's Theorem Step 2: Use the given conditions to find the value of c Answer: There ∃ c ∈ (0, 1) such that f'(c) = 0
Find the derivative of f(x) = s∈(x²) Step 1: Apply the cha∈ rule Step 2: Simplify the expression Answer: f'(x) = 2x cos(x²)

🧠 The One Thing Most Students Get Wrong

Most students get wrong the concept of continuity and differentiability at a point
They often confuse the two concepts and think that if a function is continuous at a point, then it is also differentiable at that point
However, this is not always true, and there are many examples of functions that are continuous at a point but not differentiable at that point
For example, the function f(x) = |x| is continuous at x = 0, but not differentiable at x = 0

👁️ Ayush's Note

To solve problems related to continuity and differentiability, first check the continuity of the function at the given point
Then, check if the function is differentiable at that point
Use the definition of continuity and differentiability to solve problems
Apply the cha∈ rule, quotient rule, and product rule to find the derivative of functions
Use Rolle's Theorem and Mean Value Theorem to solve problems related to continuity and differentiability

🔁 Last 5 Minutes Box

Check the continuity of the function at the given point
Check if the function is differentiable at that point
Use the definition of continuity and differentiability to solve problems
Apply the cha∈ rule, quotient rule, and product rule to find the derivative of functions
Use Rolle's Theorem and Mean Value Theorem to solve problems related to continuity and differentiability

📝 Practice MCQs

1. If f(x) is continuous on [0, 1] and differentiable on (0, 1), and f(0) = f(1), then

A) There ∃ c ∈ (0, 1) such that f'(c) = 0

B) There ∃ c ∈ (0, 1) such that f'(c) = 1

C) There ∃ c ∈ (0, 1) such that f'(c) = -1

D) There ∃ c ∈ (0, 1) such that f'(c) = 2

Answer: A) There ∃ c ∈ (0, 1) such that f'(c) = 0

2. Find the derivative of f(x) = (x² + 1)/(x + 1)

A) f'(x) = (2x(x + 1) - (x² + 1))/(x + 1)²

B) f'(x) = (2x(x + 1) + (x² + 1))/(x + 1)²

C) f'(x) = (2x(x + 1) - (x² + 1))/(x - 1)²

D) f'(x) = (2x(x + 1) + (x² + 1))/(x - 1)²

Answer: A) f'(x) = (2x(x + 1) - (x² + 1))/(x + 1)²

3. If f(x) is continuous on [a, b] and differentiable on (a, b), then

A) There ∃ c ∈ (a, b) such that f'(c) = (f(b) - f(a))/(b - a)

B) There ∃ c ∈ (a, b) such that f'(c) = (f(b) + f(a))/(b + a)

C) There ∃ c ∈ (a, b) such that f'(c) = (f(b) - f(a))/(b + a)

D) There ∃ c ∈ (a, b) such that f'(c) = (f(b) + f(a))/(b - a)

Answer: A) There ∃ c ∈ (a, b) such that f'(c) = (f(b) - f(a))/(b - a)

4. Find the derivative of f(x) = s∈(x²)

A) f'(x) = 2x cos(x²)

B) f'(x) = 2x s∈(x²)

C) f'(x) = x cos(x²)

D) f'(x) = x s∈(x²)

Answer: A) f'(x) = 2x cos(x²)

5. If f(x) is continuous at x = a then

A) f(x) is differentiable at x = a

B) f(x) is not differentiable at x = a

C) f(x) may or may not be differentiable at x = a

D) f(x) is not defined at x = a

Answer: C) f(x) may or may not be differentiable at x = a

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

📚 Related Topics

Continue your revision with these related guides:

Continuity at x = a: limₓ→an f(x) = f(a)
Differentiability at x = a: f'(a) = limₕ→₀ (f(a + h) - f(a))/h
If f(x) is differentiable at x = a then f(x) is continuous at x = a bit the converse is not always true
If f(x) and g(x) are continuous at x = a then f(x) + g(x), f(x) - g(x), f(x) × g(x), and f(x)/g(x) are continuous at x = an
If f(x) and g(x) are differentiable at x = a then f(x) + g(x), f(x) - g(x), f(x) × g(x), and f(x)/g(x) are differentiable at x = a
Cha∈ Rule: (f(g(x)))' = f'(g(x)) × g'(x)
Rolle's Theorem: if f(x) is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b), then there ∃ c ∈ (a, b) such that f'(c) = 0
Mean Value Theorem: if f(x) is continuous on [a, b] and differentiable on (a, b), then there ∃ c ∈ (a, b) such that f'(c) = (f(b) - f(a))/(b - a)