Concept Core
Limits, derivative definition, standard derivatives, and chain rule.
Limits — Standard Results
lim(x→0) sinx/x = 1 lim(x→0) (1+x)^(1/x) = e
lim(x→0) (eˣ−1)/x = 1 lim(x→0) tanx/x = 1
lim(x→0) (aˣ−1)/x = ln a lim(x→a) (xⁿ−aⁿ)/(x−a) = naⁿ⁻¹
lim(x→∞) (1+1/x)ˣ = e
These are evaluated by standard techniques: direct substitution, factorisation, L'Hôpital's rule, or standard limits above.
Derivative — Definition & First Principles
dy/dx = lim(h→0) [f(x+h) − 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[eˣ] = eˣ
d/dx[ln x] = 1/x d/dx[aˣ] = aˣ ln a
d/dx[sin x] = cos x d/dx[cos x] = −sin x
d/dx[tan x] = sec²x d/dx[sec x] = sec x tan x
d/dx[sin⁻¹x] = 1/√(1−x²) d/dx[tan⁻¹x] = 1/(1+x²)
Rules of Differentiation
Sum/Difference: d/dx[u±v] = du/dx ± dv/dx
Product: d/dx[uv] = u(dv/dx) + v(du/dx)
Quotient: d/dx[u/v] = [v(du/dx) − u(dv/dx)] / v²
Chain Rule: d/dx[f(g(x))] = f'(g(x)) × g'(x)
Applications — Tangent, Maxima, Minima
Slope of tangent to curve y=f(x) at (x₁,y₁): m = f'(x₁)
Critical points: set dy/dx = 0 → solve for x
Second derivative test: d²y/dx² > 0 → minima; d²y/dx² < 0 → maxima; = 0 → inflection (check further)
Increasing: f'(x) > 0. Decreasing: f'(x) < 0.
L'Hôpital'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[eˣ] = eˣ; d/dx[aˣ] = aˣ 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]=−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 − uv')/v²
Low d(high) minus high d(low)...
Chain Rule
d/dx[f(g)] = f'(g)·g'
Differentiate outer then inner
Inverse Trig
d/dx[sin⁻¹x] = 1/√(1−x²)
d/dx[tan⁻¹x] = 1/(1+x²)
Standard Limits
lim(x→0) sinx/x = 1
Also: (eˣ−1)/x=1; tanx/x=1 at 0
L'Hôpital's Rule
lim f/g = lim f'/g' (0/0 or ∞/∞)
Differentiate top and bottom separately
Worked Examples
5 problems — 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 × sin(3x)/(3x)
3
As x→0, 3x→0, so lim sin(3x)/(3x) = 1
✓ Limit = 3
EasyDifferentiate y = x³ − 5x² + 7x − 2▾
Find dy/dx for y = x³ − 5x² + 7x − 2.
1
Apply power rule term by term:
✓ dy/dx = 3x² − 10x + 7
MediumDifferentiate y = x² sin x using product rule▾
Find dy/dx for y = x² sin x.
1
Product rule: d/dx[u·v] = u'v + uv'
2
u = x² → u' = 2x; v = sin x → v' = cos x
3
dy/dx = 2x sin x + x² cos x
✓ dy/dx = 2x sin x + x² cos x
EAPCET LevelFind the maximum value of f(x) = 2x³ − 9x² + 12x − 2▾
Find the local maximum value of f(x) = 2x³ − 9x² + 12x − 2.
1
f'(x) = 6x² − 18x + 12 = 6(x² − 3x + 2) = 6(x−1)(x−2)
2
Critical points: x = 1, x = 2
4
At x=1: f''(1) = 12−18 = −6 < 0 → local maximum
5
At x=2: f''(2) = 24−18 = 6 > 0 → local minimum
6
Max value: f(1) = 2−9+12−2 = 3
✓ Local maximum = 3 at x = 1
Trap Questiond/dx[sin(x²)] = cos(x²) — Correct or Not?▾
A student writes d/dx[sin(x²)] = cos(x²). Is this correct?
1
The trap: Forgetting to apply the chain rule to the inner function x².
2
y = sin(u) where u = x². Chain rule: dy/dx = cos(u) × du/dx
4
dy/dx = cos(x²) × 2x = 2x cos(x²)
✓ Correct answer: 2x cos(x²) — chain rule requires multiplying by inner derivative 2x
Mistake DNA
4 differentiation errors that EAPCET distractors are built around.
🔗
Forgetting the Chain Rule
The most common differentiation error — not multiplying by the derivative of the inner function.
❌ Wrong
d/dx[e^(x²)] = e^(x²) ✗
(forgot inner derivative)
✓ Correct
d/dx[e^(x²)] = e^(x²) × 2x ✓
Outer derivative ×
Inner derivative
Chain rule: d/dx[f(g(x))] = f'(g(x)) × g'(x). The 'outer × inner' mantra: differentiate the outer function leaving inner unchanged, then multiply by derivative of inner.
➗
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[x³/x] = quotient rule
→ messy calculation ✗
✓ Correct
Simplify: x³/x = x² ✓
d/dx[x²] = 2x ✓
Algebra before calculus
Before applying any rule, simplify the expression algebraically. Products of powers → combine exponents. Rational expressions → simplify fractions. This saves significant time.
📐
L'Hôpital'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(x→1) x²/(x+1):
Apply L'Hôpital: 2x/1 = 2 ✗
(not 0/0; direct sub: 1/2)
✓ Correct
Check: at x=1: 1/2 ✓
Not 0/0, so direct sub
works → answer 1/2 ✓
Always check the form of the limit first. Direct substitution → not indeterminate → use direct substitution. Only 0/0 or ∞/∞ → L'Hôpital.
🎯
d²y/dx² > 0 → Maximum (Confusing with Minimum)
Second derivative test: f''(x) > 0 at critical point → concave up → minimum. f''(x) < 0 → concave down → maximum.
❌ Wrong
f''(x) > 0 at critical point
→ maximum ✗
(common sign reversal)
✓ Correct
f''(x) > 0 → minimum ✓
f''(x) < 0 → maximum ✓
'Smiley face' (+) = minimum
'Frown' (−) = maximum
Memory: concave up (∪ shape) = minimum (like a valley). Concave down (∩ shape) = maximum (like a hill). d²y/dx² > 0 gives upward concavity = valley = minimum.
Chapter Intelligence
Calculus is the highest-scoring domain in EAPCET Maths — invest time here for maximum returns.
EAPCET Weightage (2019–2024)
Derivatives of standard functions~9 Chain rule applications~8 Limits using standard results~5 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 × trigTangent slope at given pointContinuity check at a pointImplicit differentiation
Exam Strategy
- Chain rule is needed for ANY composite function — sin(2x), e^(x²), ln(x²+1). The outer derivative × inner derivative mantra never fails.
- Limits: try direct substitution first. If 0/0 or ∞/∞ forms appear → factorize/simplify/L'Hôpital. If sin(ax)/x form → multiply and divide by a.
- Max/min: find f'(x) = 0 for critical points; apply f''(x) test. f'' < 0 → max; f'' > 0 → min.
- Differentiation and Integration together account for 10–12 EAPCET marks. This is the single most important section to master.