CBSE Class 12 · Mathematics
Continuity and Differentiability
Official NCERT chapter from Mathematics Part I–II (book code lemh1). ExamMaster notes are original teaching at CBSE Class 12 depth.
This lesson follows the official chapter “Continuity and Differentiability” in Mathematics Part I–II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.
- CBSE Class 12
- Medium level
- 8 concepts
1Continuity
Continuity at a: lim_{x→a} f(x) = f(a) as taught (both sides if needed). A hole or a jump fails. f(x)=x² is continuous at 2 because the limit is 4 and f(2)=4. Continuity is a limit-equals-value test, not “smooth looking”.
A removable hole with f(a) undefined is not continuous at a.
Figure. Continuity at a: the curve meets the plotted value f(a). Left and right approach the same height the function writes at a.
How it works
- Compute the limit and f(a)The two sides.
- Ask whether they matchContinuous if yes.
- Check left and right if the lesson split themHonest.
2Differentiability
Differentiability: the limit of [f(a+h)−f(a)]/h exists as taught. Differentiable implies continuous; the converse can fail (a corner). f(x)=|x| at 0 is the school continuous-not-diff case if taught. Differentiable is a slope-limit, not “drawn without lifting”.
A sharp corner called differentiable because it is continuous is a miss.
Figure. The V is continuous at 0 — both sides meet on the axis — but the corner has no single tangent, so |x| is not differentiable there. y is never negative, so the x-axis stays on the frame; x takes both signs, so the y-axis sits mid-figure.
How it works
- Form the difference quotient and take h→0The test.
- If it exists, f is differentiable thereThe word.
- Keep continuous-not-diff as a named cautionA corner.
3Exponential and Logarithmic Functions
Exponential and log (school): d/dx(e^x)=e^x; d/dx(ln x)=1/x as taught. These are new derivative facts, not a second continuity test. Domain x>0 for ln as taught.
d/dx(e^x)=x e^{x−1} is a power-rule steal.
Figure. e^x passes through (0,1) and is drawn only while it stays on the frame. ln x is its inverse: it meets the x-axis at (1,0) and is the reflection of e^x in the dashed line y=x.
How it works
- Name e^x or ln xThe function.
- Write the taught derivativeThe fact.
- Keep ln on x>0The domain.
4Logarithmic Differentiation
Logarithmic differentiation: take ln of both sides when a product/power mess as taught, then differentiate. y=x^x is the school case if named. It is a method, not a new function-family.
Taking ln and then forgetting to multiply by y at the end is a miss.
Figure. Logarithmic differentiation: take ln of a product (or a power) first, differentiate, then multiply back by y. The sum of relative rates is the point.
How it works
- Take ln, then d/dxThe method.
- Bring y back if you used ln yThe chain.
- Keep it for a messy product/powerThe use.
5Derivatives of Functions in Parametric Forms
Parametric: x=x(t), y=y(t), then dy/dx = (dy/dt)/(dx/dt) as taught when dx/dt≠0. Parametric is two-rules-one-slope, not “x and y forgot each other”. x=t², y=t³ → dy/dx = 3t/2 (t≠0).
Adding dy/dt and dx/dt as the slope is a miss.
Figure. At t=1 the parametric path is (2,1) and dy/dx = (dy/dt)/(dx/dt) = 2t/2 = t = 1. The dashed tangent is the line y=x-1.
How it works
- Differentiate both with tThe two rates.
- Divide dy/dt by dx/dtThe slope.
- Refuse the divide if dx/dt=0A vertical caution.
6Second Order Derivative
Second order: d²y/dx² is the derivative of dy/dx as taught. For y=x³, y′=3x², y″=6x. Second order is a second slope-of-slope, not a double of y′.
Writing y″= (y′)² is a miss.
Figure. f''=2>0: the parabola stays above its tangent. The second derivative is the concavity test, not a second copy of the slope.
How it works
- Differentiate y′ once morey″.
- Keep it d/dx of the first derivativeThe meaning.
- Refuse (y′)² as this headingDifferent write.
7A definition is a test you can run
Continuity / derivative-word is a test: limit=value, a difference-quotient, e^x/ln, a parametric divide, or y″. If you only say “smooth”, you have a heading.
A graph-sticker is not the test.
Figure. Continuity is a test you can run: left-hand limit, right-hand limit, and the written value f(a) must be the same number.
How it works
- Name the test or the derivative-factThe object.
- Give the school sentenceThe test.
- Then the word has contentThe definition ran.
8Name the given before the unknown
The given is a rule and a point. The unknown is continuous? or the derivative. Copy f(2)=4 before you claim a hole at 2.
Using left-limit only when sides differ is a silent skip.
Figure. Name the given function before the unknown derivative. The rule you pick (product, chain, parametric) waits until f is written.
How it works
- Copy the rule and the pointThe given.
- Name continuity or f′The unknown.
- Then computeGiven first.
d/dx(e^x) is
- e^x
- x e^{x−1}
- 1/x
The exponential fact.
Notes
- The official chapter title is “Continuity and Differentiability”. Teach the school test for that title, not a contest shortcut.
- If a step needs a later class, stop. The next official chapter will pick it up.
Formulas
- continuous: lim f = f(a)
- dy/dx = (dy/dt)/(dx/dt)
- d/dx(e^x)=e^x, d/dx(ln x)=1/x
Recap
Hold these pegs from the official chapter “Continuity and Differentiability”. The wording is ExamMaster’s teaching, not a textbook recap.
- Continuity
- Continuity at a: lim_{x→a} f(x) = f(a) as taught (both sides if needed).
- Differentiability
- Differentiability: the limit of [f(a+h)−f(a)]/h exists as taught.
- Exponential and Logarithmic Functions
- Exponential and log (school): d/dx(e^x)=e^x; d/dx(ln x)=1/x as taught.
- Logarithmic Differentiation
- Logarithmic differentiation: take ln of both sides when a product/power mess as taught, then differentiate.
- Derivatives of Functions in Parametric Forms
- Parametric: x=x(t), y=y(t), then dy/dx = (dy/dt)/(dx/dt) as taught when dx/dt≠0.
- Second Order Derivative
- Second order: d²y/dx² is the derivative of dy/dx as taught.
Practise Continuity and Differentiability
Reading is free and needs no account. Practice, mocks and progress live in the app.
- A 4-question practice set that ends the chapter
- 1 quick check with worked explanations
- Timed mocks scored with the real marking scheme
- Readiness tracked per topic, kept on your device