Chapter Hub
Continuity and Differentiability
This chapter connects limits with the behaviour of functions near a point and then builds the derivative as a precise rate of change. Continuity checks whether the function value agrees with the limiting value, while differentiability checks whether the left-hand and right-hand slopes agree. A major exam skill in this chapter is knowing which rule applies under which condition. For example, differentiability implies continuity, but continuity alone does not imply differentiability. Similarly, chain rule, implicit differentiation, logarithmic differentiation, and parametric differentiation each have a specific structure to identify before starting calculation. The chapter is formula-rich, but marks are usually lost in conditions and algebra. You must state one-sided limits, left and right derivatives, domains of inverse trigonometric and logarithmic functions, and non-zero denominator conditions wherever needed. Most long-answer questions combine two or more methods, such as continuity with differentiability, chain rule with inverse trigonometric functions, or logarithmic differentiation with product and quotient forms.
Difficulty
Medium
Study time
70-90 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 70 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
Start with one of the buckets below, then open the full map when you want the complete concept roadmap.
Key Concepts
Work through the concepts in order. Each one has an explanation, a worked example, a self-check and an exam answer.
All Concepts
All 7 concepts of the chapter, in the order the NCERT chapter teaches them.
Continuity at a Point
A function f is continuous at x = a if f(a) is defined, lim(x → a) f(x) exists, and lim(x → a) f(x) = f(a).
Differentiability and Continuity
A function f is differentiable at x = a if its left-hand derivative and right-hand derivative at a both exist and are equal.
Chain Rule for Composite Functions
The chain rule gives the derivative of a composite function f(g(x)) by differentiating the outer function at g(x) and multiplying by the derivative of g(x).
Implicit Differentiation
Implicit differentiation is used when x and y are related by an equation and y is not first written explicitly as a function of x.
Derivatives of Inverse Trigonometric, Exponential, and Logarithmic Functions
This concept covers standard derivatives of inverse trigonometric functions, exponential functions, and logarithmic functions, along with their domain conditions.
Logarithmic Differentiation
Logarithmic differentiation is a method where logarithms are taken on both sides before differentiating, especially for variable powers and complicated products or quotients.
Parametric and Second-Order Derivatives
For parametric equations x = f(t), y = g(t), the derivative dy/dx is found as (dy/dt)/(dx/dt), provided dx/dt is not zero. The second-order derivative measures the rate of change of dy/dx with respect to x.
Exam Intelligence
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
Common Traps
- Assuming continuity proves differentiability.
- Using the wrong branch to calculate f(a) in a piecewise function.
- Forgetting the inner derivative in chain rule.
- Differentiating yn as nyn−1 instead of nyn−1dy/dx during implicit differentiation.
- Missing the negative sign in the derivative of cos−1 x.
- Forgetting to multiply by y at the end of logarithmic differentiation.
- Calling d/dt(dy/dx) the second derivative without dividing by dx/dt.
How CBSE Asks This Chapter
- Continuity at a Point
Usually asked by giving a piecewise function and asking for unknown constants that make it continuous at one or two points.
- Differentiability and Continuity
Often asked as a parameter-finding problem where a piecewise function must be continuous and differentiable at the joining point.
- Chain Rule for Composite Functions
Appears as direct differentiation, part of implicit differentiation, or combined with inverse trigonometric, logarithmic, and exponential functions.
- Implicit Differentiation
Questions give equations such as x2 + y2 = a2, xy = c, or expressions with powers and products, then ask for dy/dx.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Remember
Continuity at x = a: LHL = RHL = f(a).Trap to avoid
Students often equate only the two formula pieces and forget to compare them with f(a), especially when the equality sign belongs to one side of a piecewise function.
Differentiability and Continuity
Remember
LHD = lim(h → 0-) [f(a + h) - f(a)]/h and RHD = lim(h → 0+) [f(a + h) - f(a)]/h.Trap to avoid
A frequent wrong claim is that continuity automatically proves differentiability.
Chain Rule for Composite Functions
Remember
If y = f(g(x)), then dy/dx = f'(g(x))g'(x).Trap to avoid
Students differentiate the outer function correctly but forget to multiply by the derivative of the inner function.
Remember
Key condition: y is treated as a differentiable function of x.Trap to avoid
Students differentiate y2 as 2y instead of 2y dy/dx, which ignores that y depends on x.
Derivatives of Inverse Trigonometric, Exponential, and Logarithmic Functions
Remember
d/dx(sin−1 x) = 1/√(1 - x2), d/dx(cos−1 x) = -1/√(1 - x2), d/dx(tan−1 x) = 1/(1 + x2), d/dx(ex) = ex, d/dx(ax) = ax log a, d/dx(log x) = 1/x for x > 0.Trap to avoid
Students often write the correct outer derivative but ignore the derivative of the inside expression, especially in sin−1(2x) or log(3x + 1).
Remember
If y = u(x)v(x) with u(x) > 0, then log y = v(x)log u(x), and dy/dx = y[v'(x)log u(x) + v(x)u'(x)/u(x)].Trap to avoid
Students take logarithms but forget to multiply the final derivative expression by y when converting from (1/y)dy/dx to dy/dx.
Parametric and Second-Order Derivatives
Remember
For x = f(t), y = g(t): dy/dx = (dy/dt)/(dx/dt), where dx/dt ≠ 0.Trap to avoid
Students calculate d/dt(dy/dx) and call it d2y/dx2 without dividing by dx/dt.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Continuity at a Point | Continuity at x = a: LHL = RHL = f(a). | Students often equate only the two formula pieces and forget to compare them with f(a), especially when the equality sign belongs to one side of a piecewise function. |
| Differentiability and Continuity | LHD = lim(h → 0-) [f(a + h) - f(a)]/h and RHD = lim(h → 0+) [f(a + h) - f(a)]/h. | A frequent wrong claim is that continuity automatically proves differentiability. |
| Chain Rule for Composite Functions | If y = f(g(x)), then dy/dx = f'(g(x))g'(x). | Students differentiate the outer function correctly but forget to multiply by the derivative of the inner function. |
| Implicit Differentiation | Key condition: y is treated as a differentiable function of x. | Students differentiate y2 as 2y instead of 2y dy/dx, which ignores that y depends on x. |
| Derivatives of Inverse Trigonometric, Exponential, and Logarithmic Functions | d/dx(sin−1 x) = 1/√(1 - x2), d/dx(cos−1 x) = -1/√(1 - x2), d/dx(tan−1 x) = 1/(1 + x2), d/dx(ex) = ex, d/dx(ax) = ax log a, d/dx(log x) = 1/x for x > 0. | Students often write the correct outer derivative but ignore the derivative of the inside expression, especially in sin−1(2x) or log(3x + 1). |
| Logarithmic Differentiation | If y = u(x)v(x) with u(x) > 0, then log y = v(x)log u(x), and dy/dx = y[v'(x)log u(x) + v(x)u'(x)/u(x)]. | Students take logarithms but forget to multiply the final derivative expression by y when converting from (1/y)dy/dx to dy/dx. |
| Parametric and Second-Order Derivatives | For x = f(t), y = g(t): dy/dx = (dy/dt)/(dx/dt), where dx/dt ≠ 0. | Students calculate d/dt(dy/dx) and call it d2y/dx2 without dividing by dx/dt. |
Practice
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