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Mathematics → Limits & Differentiation
TG EAPCET 2027
Chapter progress
Layer 1 of 5
MathematicsVery High Weightage β˜…β˜…β˜…β˜…β˜…Class 11 + 12

Limits & Differentiation

Limits, continuity, derivatives, and their applications β€” one of the two biggest calculus chapters in EAPCET Maths. Expect 6–8 questions across differentiation and integration.

4–5Questions in EAPCET
~5%Paper Weightage
15Core Formulas
4Mistake Traps

Concept Core

Limits, derivative definition, standard derivatives, and chain rule.

Limits β€” Standard Results
lim(xβ†’0) sinx/x = 1             lim(xβ†’0) (1+x)^(1/x) = e lim(xβ†’0) (eΛ£βˆ’1)/x = 1         lim(xβ†’0) tanx/x = 1 lim(xβ†’0) (aΛ£βˆ’1)/x = ln a      lim(xβ†’a) (xβΏβˆ’aⁿ)/(xβˆ’a) = naⁿ⁻¹ lim(xβ†’βˆž) (1+1/x)Λ£ = e

These are evaluated by standard techniques: direct substitution, factorisation, L'HΓ΄pital's rule, or standard limits above.

Derivative β€” Definition & First Principles
dy/dx = lim(hβ†’0) [f(x+h) βˆ’ f(x)] / h

This is the first-principles (ab initio) definition. For EAPCET, learn to apply standard derivative formulas directly rather than computing from first principles each time.

Standard Derivatives
d/dx[xⁿ] = nxⁿ⁻¹          d/dx[eΛ£] = eΛ£ d/dx[ln x] = 1/x          d/dx[aΛ£] = aΛ£ ln a d/dx[sin x] = cos x      d/dx[cos x] = βˆ’sin x d/dx[tan x] = secΒ²x     d/dx[sec x] = sec x tan x d/dx[sin⁻¹x] = 1/√(1βˆ’xΒ²)   d/dx[tan⁻¹x] = 1/(1+xΒ²)
Rules of Differentiation

Sum/Difference: d/dx[uΒ±v] = du/dx Β± dv/dx

Product: d/dx[uv] = u(dv/dx) + v(du/dx)

Quotient: d/dx[u/v] = [v(du/dx) βˆ’ u(dv/dx)] / vΒ²

Chain Rule: d/dx[f(g(x))] = f'(g(x)) Γ— g'(x)

Applications β€” Tangent, Maxima, Minima

Slope of tangent to curve y=f(x) at (x₁,y₁): m = f'(x₁)

Critical points: set dy/dx = 0 β†’ solve for x

Second derivative test: dΒ²y/dxΒ² > 0 β†’ minima; dΒ²y/dxΒ² < 0 β†’ maxima; = 0 β†’ inflection (check further)

Increasing: f'(x) > 0. Decreasing: f'(x) < 0.

L'HΓ΄pital's Rule

For 0/0 or ∞/∞ indeterminate forms:

lim f(x)/g(x) = lim f'(x)/g'(x)

Differentiate numerator and denominator separately (NOT the quotient rule). Apply repeatedly if still indeterminate.

Verify it's truly 0/0 or ∞/∞ before applying L'Hôpital's rule.

Formula Vault

All derivative formulas and rules for EAPCET.

Power Rule
d/dx[xⁿ] = nxⁿ⁻¹
Works for all real n
Exponential
d/dx[eΛ£] = eΛ£; d/dx[aΛ£] = aΛ£ ln a
e is unique: self-derivative
Logarithm
d/dx[ln x] = 1/x
x > 0 required
Trig Derivatives
d/dx[sin x]=cosx; d/dx[cos x]=βˆ’sinx
Chain rule for composite trig
Product Rule
d/dx[uv] = u'v + uv'
Both terms positive
Quotient Rule
d/dx[u/v] = (u'v βˆ’ uv')/vΒ²
Low d(high) minus high d(low)...
Chain Rule
d/dx[f(g)] = f'(g)Β·g'
Differentiate outer then inner
Inverse Trig
d/dx[sin⁻¹x] = 1/√(1βˆ’xΒ²)
d/dx[tan⁻¹x] = 1/(1+x²)
Standard Limits
lim(x→0) sinx/x = 1
Also: (eΛ£βˆ’1)/x=1; tanx/x=1 at 0
L'HΓ΄pital's Rule
lim f/g = lim f'/g' (0/0 or ∞/∞)
Differentiate top and bottom separately

Worked Examples

5 problems β€” limits, chain rule, product rule, maxima, and a classic trap.

EasyEvaluate lim(x→0) sin(3x)/x▾
Evaluate: lim(x→0) sin(3x)/x.
1
Rewrite to match standard form lim sin(u)/u = 1:
2
sin(3x)/x = 3 Γ— sin(3x)/(3x)
3
As x→0, 3x→0, so lim sin(3x)/(3x) = 1
4
Answer: 3 Γ— 1 = 3
βœ“  Limit = 3
EasyDifferentiate y = xΒ³ βˆ’ 5xΒ² + 7x βˆ’ 2β–Ύ
Find dy/dx for y = xΒ³ βˆ’ 5xΒ² + 7x βˆ’ 2.
1
Apply power rule term by term:
2
dy/dx = 3xΒ² βˆ’ 10x + 7
βœ“  dy/dx = 3xΒ² βˆ’ 10x + 7
MediumDifferentiate y = xΒ² sin x using product ruleβ–Ύ
Find dy/dx for y = xΒ² sin x.
1
Product rule: d/dx[uΒ·v] = u'v + uv'
2
u = xΒ² β†’ u' = 2x; v = sin x β†’ v' = cos x
3
dy/dx = 2x sin x + xΒ² cos x
βœ“  dy/dx = 2x sin x + xΒ² cos x
EAPCET LevelFind the maximum value of f(x) = 2xΒ³ βˆ’ 9xΒ² + 12x βˆ’ 2β–Ύ
Find the local maximum value of f(x) = 2xΒ³ βˆ’ 9xΒ² + 12x βˆ’ 2.
1
f'(x) = 6xΒ² βˆ’ 18x + 12 = 6(xΒ² βˆ’ 3x + 2) = 6(xβˆ’1)(xβˆ’2)
2
Critical points: x = 1, x = 2
3
f''(x) = 12x βˆ’ 18
4
At x=1: f''(1) = 12βˆ’18 = βˆ’6 < 0 β†’ local maximum
5
At x=2: f''(2) = 24βˆ’18 = 6 > 0 β†’ local minimum
6
Max value: f(1) = 2βˆ’9+12βˆ’2 = 3
βœ“  Local maximum = 3 at x = 1
Trap Questiond/dx[sin(xΒ²)] = cos(xΒ²) β€” Correct or Not?β–Ύ
A student writes d/dx[sin(xΒ²)] = cos(xΒ²). Is this correct?
1
The trap: Forgetting to apply the chain rule to the inner function xΒ².
2
y = sin(u) where u = xΒ². Chain rule: dy/dx = cos(u) Γ— du/dx
3
du/dx = d/dx[xΒ²] = 2x
4
dy/dx = cos(xΒ²) Γ— 2x = 2x cos(xΒ²)
βœ“  Correct answer: 2x cos(xΒ²) β€” chain rule requires multiplying by inner derivative 2x

Mistake DNA

4 differentiation errors that EAPCET distractors are built around.

πŸ”—
Forgetting the Chain Rule
The most common differentiation error β€” not multiplying by the derivative of the inner function.
❌ Wrong
d/dx[e^(xΒ²)] = e^(xΒ²) βœ— (forgot inner derivative)
βœ“ Correct
d/dx[e^(xΒ²)] = e^(xΒ²) Γ— 2x βœ“ Outer derivative Γ— Inner derivative
Chain rule: d/dx[f(g(x))] = f'(g(x)) Γ— g'(x). The 'outer Γ— inner' mantra: differentiate the outer function leaving inner unchanged, then multiply by derivative of inner.
βž—
Using Quotient Rule When Product Rule is Simpler
d/dx[x²/√x] = d/dx[x^(3/2)] using algebra is much faster than quotient rule. Always simplify first.
❌ Wrong
d/dx[xΒ³/x] = quotient rule β†’ messy calculation βœ—
βœ“ Correct
Simplify: xΒ³/x = xΒ² βœ“ d/dx[xΒ²] = 2x βœ“ Algebra before calculus
Before applying any rule, simplify the expression algebraically. Products of powers β†’ combine exponents. Rational expressions β†’ simplify fractions. This saves significant time.
πŸ“
L'HΓ΄pital's Rule Applied to Non-Indeterminate Forms
L'Hôpital's rule only applies to 0/0 or ∞/∞ forms. Applying it to other limits gives wrong answers.
❌ Wrong
lim(xβ†’1) xΒ²/(x+1): Apply L'HΓ΄pital: 2x/1 = 2 βœ— (not 0/0; direct sub: 1/2)
βœ“ Correct
Check: at x=1: 1/2 βœ“ Not 0/0, so direct sub works β†’ answer 1/2 βœ“
Always check the form of the limit first. Direct substitution β†’ not indeterminate β†’ use direct substitution. Only 0/0 or ∞/∞ β†’ L'HΓ΄pital.
🎯
dΒ²y/dxΒ² > 0 β†’ Maximum (Confusing with Minimum)
Second derivative test: f''(x) > 0 at critical point β†’ concave up β†’ minimum. f''(x) < 0 β†’ concave down β†’ maximum.
❌ Wrong
f''(x) > 0 at critical point β†’ maximum βœ— (common sign reversal)
βœ“ Correct
f''(x) > 0 β†’ minimum βœ“ f''(x) < 0 β†’ maximum βœ“ 'Smiley face' (+) = minimum 'Frown' (βˆ’) = maximum
Memory: concave up (βˆͺ shape) = minimum (like a valley). Concave down (∩ shape) = maximum (like a hill). dΒ²y/dxΒ² > 0 gives upward concavity = valley = minimum.

Chapter Intelligence

Calculus is the highest-scoring domain in EAPCET Maths β€” invest time here for maximum returns.

EAPCET Weightage (2019–2024)
Derivatives of standard functions
~9
Chain rule applications
~8
Maxima/minima problems
~6
Limits using standard results
~5
L'HΓ΄pital's rule
~4
Rolle's & MVT
~3
High-Yield PYQ Patterns
d/dx of sin(f(x)), e^(f(x))Find maxima/minima from f''Evaluate limit: sin(ax)/x formProduct rule: x^n Γ— trigTangent slope at given pointContinuity check at a pointImplicit differentiation
Exam Strategy
  • Chain rule is needed for ANY composite function β€” sin(2x), e^(xΒ²), ln(xΒ²+1). The outer derivative Γ— inner derivative mantra never fails.
  • Limits: try direct substitution first. If 0/0 or ∞/∞ forms appear β†’ factorize/simplify/L'HΓ΄pital. If sin(ax)/x form β†’ multiply and divide by a.
  • Max/min: find f'(x) = 0 for critical points; apply f''(x) test. f'' < 0 β†’ max; f'' > 0 β†’ min.
  • Differentiation and Integration together account for 10–12 EAPCET marks. This is the single most important section to master.
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