Differentiation

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Differentiation — Key Facts for JEE Main Derivative: rate of change of function with respect to variable; f’(x) = dy/dx = lim(h→0) [f(x+h) − f(x)]/h dy/dx represents slope of tangent to curve at point (x, y) Second derivative: d²y/dx² = derivative of dy/dx Standard derivatives: d/dx(x^n) = nx^{n−1}; d/dx(sin x) = cos x; d/dx(e^x) = e^x; d/dx(ln x) = 1/x Chain rule: d/dx[f(g(x))] = f’(g(x)) · g’(x) ⚡ Exam tip: JEE Main tests chain rule heavily — always identify the outer and inner functions before differentiating!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Differentiation — JEE Main Study Guide

Basic differentiation formulas:

• d/dx(x^n) = nx^{n−1} (power rule)
• d/dx(constant) = 0
• d/dx(e^x) = e^x
• d/dx(a^x) = a^x · ln a
• d/dx(ln x) = 1/x
• d/dx(log_a x) = 1/(x · ln a)

Trigonometric derivatives:

• d/dx(sin x) = cos x
• d/dx(cos x) = −sin x
• d/dx(tan x) = sec²x
• d/dx(cot x) = −cosec²x
• d/dx(sec x) = sec x tan x
• d/dx(cosec x) = −cosec x cot x

Inverse trig derivatives:

• d/dx(sin⁻¹x) = 1/√(1−x²)
• d/dx(cos⁻¹x) = −1/√(1−x²)
• d/dx(tan⁻¹x) = 1/(1+x²)
• d/dx(cot⁻¹x) = −1/(1+x²)

Product rule: d/dx(f·g) = f’·g + f·g’ Quotient rule: d/dx(f/g) = (f’·g − f·g’)/g² Chain rule: d/dx[f(g(x))] = f’(g(x)) · g’(x)

Parametric differentiation: If x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt) = g’(t)/f’(t)

Second order derivative: d²y/dx² = d/dx(dy/dx); for parametric: d²y/dx² = d/dx(dy/dx)/(dx/dt)

Implicit differentiation: Differentiate both sides with respect to x, treating y as function of x For terms with y, use chain rule: d/dx(y²) = 2y · dy/dx

Logarithmic differentiation: Take ln of both sides for products/quotients with powers Example: y = x^x → ln y = x ln x → dy/dx = y(ln x + 1)

Inverse function derivative: If y = f(x), then dx/dy = 1/(dy/dx) = 1/f’(x)

• Key formula: d/dx(x^n) = nx^{n−1}; chain rule: d/dx[f(g(x))] = f’(g(x))·g’(x)
• Common trap: When differentiating sin(x²), the inner function x² must also be differentiated: d/dx[sin(x²)] = cos(x²) · 2x
• Exam weight: 2–3 questions per year (8–12 marks); foundational for integration and applications

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Differentiation — Comprehensive JEE Main Notes

Successive differentiation (nth derivative):

• d^n/dx^n (x^m) = m(m−1)…(m−n+1) · x^{m−n}
• d^n/dx^n (sin(ax+b)) = a^n sin(ax+b + nπ/2)
• d^n/dx^n (e^{ax}) = a^n e^{ax}
• d^n/dx^n (ln(ax+b)) = (−1)^{n−1} · (n−1)! · a^n/(ax+b)^n

Leibniz theorem for nth derivative of product: If y = u·v, then d^n/dx^n (uv) = Σ_{k=0}^{n} C(n,k) · u^{(n−k)} · v^{(k)} Used for products where both functions have simple nth derivatives

Mean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that: f’(c) = [f(b) − f(a)]/(b − a) Geometrically: tangent at c is parallel to chord AB

Rolle’s Theorem: If f(a) = f(b), then there exists c ∈ (a, b) where f’(c) = 0 Special case of MVT

Taylor and Maclaurin series: f(x) = f(a) + f’(a)(x−a)/1! + f”(a)(x−a)²/2! + … + f^{(n)}(a)(x−a)^n/n! + R_n At a = 0: Maclaurin series for e^x, sin x, cos x, ln(1+x)

L’Hôpital’s rule for differentiation: For indeterminate 0/0 or ∞/∞: lim f(x)/g(x) = lim f’(x)/g’(x) if RHS limit exists Can be applied repeatedly

Differentiation under integral (Leibniz rule): d/dx [∫{a(x)}^{b(x)} f(x,t) dt] = f(x, b(x))·b’(x) − f(x, a(x))·a’(x) + ∫{a(x)}^{b(x)} ∂f/∂x · dt

Maxima and minima: First derivative test: f’(x) = 0 → check f”(x):

• f”(x) > 0: local minimum
• f”(x) < 0: local maximum Second derivative test fails when f”(x) = 0 → use first derivative sign change test

Function increasing/decreasing:

• f’(x) > 0 on interval → f is increasing
• f’(x) < 0 on interval → f is decreasing
• Strictly monotonic: f’(x) ≥ 0 (or ≤ 0) and f’(x) = 0 only at isolated points

Envelope and orthogonal trajectories: For family of curves y = f(x, c), eliminate c between y = f(x,c) and f_x(x,c) = 0 to get envelope

Derivative of determinant: If D(x) = |a(x) b(x); c(x) d(x)|, then D’(x) = |a’(x) b(x); c’(x) d(x)| + |a(x) b’(x); c(x) d’(x)|

Differential of arc length: ds = √[(dx)² + (dy)²]; for parametric: ds = √[(dx/dt)² + (dy/dt)²] dt

• Remember: Chain rule is the most important — always identify outer and inner function; d/dx[f(g(x))] = f’(g(x))·g’(x); for implicit differentiation, treat y as function of x and use chain rule on y terms
• Previous years: “Find derivative of e^{x²}·sin x using chain and product rule” [2023]; “If y = x^{x}, find dy/dx” [2024]; “Find d²y/dx² if x = a cos θ, y = b sin θ” [2024]

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📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Time	3 hours
Marks	300 (90 per subject)
Section	Physics (30), Chemistry (30), Mathematics (30)
Negative	−1 for wrong answer
Mode	Computer-based

🎯 High-Yield Topics for JEE Main Mathematics

• Calculus (Differentiation + Integration) — ~35 marks combined
• Coordinate Geometry (straight lines, circles, conics) — ~20 marks
• Algebra (Complex Numbers, Quadratics, P&C, Probability) — ~25 marks
• Trigonometry + Inverse Trigonometry — ~15 marks
• Vector + 3D — ~15 marks

📝 Previous Year Question Patterns

• Differentiation: 2–3 questions per year, 8–12 marks
• Common patterns: chain rule application, differentiation of inverse functions, parametric differentiation, logarithmic differentiation, nth derivative
• Weight: very high frequency, foundational for entire calculus

💡 Pro Tips

• Always identify outer and inner function before applying chain rule
• For implicit differentiation, differentiate each term treating y as function of x
• For y = f(x)^g(x), use logarithmic differentiation: ln y = g(x) · ln f(x)
• Remember the derivative of e^x is e^x, and d/dx(ln x) = 1/x
• For inverse trig derivatives, memorize the domain restrictions
• Practice nth derivative problems — they appear occasionally and are scoring if you know the pattern
• For maximum/minimum, always check second derivative or first derivative sign change

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

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