Questions
5–7 questions per paper
Difficulty
Medium
Importance
Core — never skip
Overview
Continuity and Differentiability form the bedrock of Calculus, bridgeing the gap between limits and integration. Mastery of these concepts is essential for scoring in CBSE board exams as they account for significant marks and provide the groundwork for Application of Derivatives.
Continuity of a Function
A function is continuous at a point if the limit exists and equals the functional value. For board exams, you must be comfortable checking the Left Hand Limit (LHL), Right Hand Limit (RHL), and the value at the point to determine continuity over an interval.
- LHL = RHL = f(a) for continuity at x=a
- Polynomial functions are continuous everywhere
- Rational functions are continuous except where the denominator is zero
- Intermediate Value Theorem application
- Algebra of continuous functions: f+g, f-g, f*g are continuous
Chain Rule and Derivatives
The Chain Rule is the primary tool for differentiating composite functions. It is frequently tested in problems involving trigonometric, exponential, and logarithmic functions nested within one another.
- d/dx [f(g(x))] = f'(g(x)) * g'(x)
- d/dx (sin x) = cos x
- d/dx (e^x) = e^x
- d/dx (log |x|) = 1/x
- Product Rule: d/dx (uv) = u'v + uv'
Implicit and Logarithmic Differentiation
Implicit differentiation is used when variables x and y cannot be easily separated. Logarithmic differentiation simplifies complex power-based functions by transforming exponents into products.
- Treat y as a function of x when differentiating implicitly
- Use log differentiation for functions of form [f(x)]^g(x)
- d/dx (log y) = (1/y) * (dy/dx)
- Differentiate term by term for implicit equations
- Apply chain rule strictly for dy/dx terms
Formula Sheet
f(a) = lim(x->a-) f(x) = lim(x->a+) f(x)
d/dx [f(g(x))] = f'(g(x)) * g'(x)
d/dx [u/v] = (u'v - uv') / v^2
d/dx [x^n] = n * x^(n-1)
d/dx [a^x] = a^x * log(a)
d/dx [log_a(x)] = 1 / (x * log(a))
Exam Tip
Always write the limit conditions (LHL, RHL, f(a)) explicitly in your answer script, as step-marking is strictly followed for continuity proofs.
Common Mistakes
- Forgetting to include the dy/dx term when differentiating y implicitly
- Failing to check LHL and RHL separately in piecewise functions
- Ignoring the modulus sign or absolute value constraints in logarithmic differentiation
More Revision Notes
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