Calculus Formula Sheet - IIT-JEE MAINS 2027

Welcome to the ultimate LibreTexts-style reference guide for Calculus, meticulously engineered for IIT-JEE MAINS 2027. Differential and Integral Calculus constitute approximately 35% to 40% of the total marks in the JEE Mathematics paper. Mastering foundational limits, standard derivatives, and integration strategies is crucial for securing a high percentile under the National Testing Agency (NTA) exam pattern.

1. Limits and Continuity

Limits form the backbone of calculus. In the IIT-JEE Mains syllabus, limit evaluation tests both trigonometric identities and algebraic expansion techniques.

Standard Limits

Memorize these standard indeterminate limits to solve questions rapidly during the exam:

• Trigonometric Limits: \(\lim_{x \to 0} \frac{\sin x}{x} = 1\), \(\lim_{x \to 0} \frac{\tan x}{x} = 1\)
• Cosine Forms: \(\lim_{x \to 0} \frac{1 - \cos x}{x} = 0\), \(\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}\)
• Exponential & Logarithmic: \(\lim_{x \to 0} \frac{e^x - 1}{x} = 1\), \(\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a\), \(\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1\)
• Algebraic Limit: \(\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}\)
• Asymptotic Exponential Form: \(\lim_{x \to \infty} \bigl(1 + \frac{k}{x}\bigr)^x = e^k\)

L'Hôpital's Rule

When direct substitution into \(\lim_{x \to c} \frac{f(x)}{g(x)}\) yields indeterminate forms such as \(\bigl[\frac{0}{0}\bigr]\) or \(\bigl[\frac{\infty}{\infty}\bigr]\), differentiate the numerator and denominator independently:

\[\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\]

2. Derivatives and Differentiation Rules

Differentiation measures the instantaneous rate of change of functions and is tested heavily in both direct differential calculus and Application of Derivatives (AOD).

Fundamental Differentiation Rules

• Product Rule: \((uv)' = u'v + uv'\)
• Quotient Rule: \(\bigl(\frac{u}{v}\bigr)' = \frac{u'v - uv'}{v^2}\)
• Chain Rule: \(\frac{d}{dx}\bigl[f(g(x))\bigr] = f'(g(x)) \cdot g'(x)\)

Common Derivatives Table

Below is a summary of standard functions and their corresponding derivatives:

• Algebraic: \(\frac{d}{dx}(x^n) = n x^{n-1}\)
• Exponential: \(\frac{d}{dx}(e^x) = e^x\), \(\frac{d}{dx}(a^x) = a^x \ln a\)
• Logarithmic: \(\frac{d}{dx}(\ln |x|) = \frac{1}{x}\)
• Trigonometric Functions:
  • \(\frac{d}{dx}(\sin x) = \cos x\)
  • \(\frac{d}{dx}(\cos x) = -\sin x\)
  • \(\frac{d}{dx}(\tan x) = \sec^2 x\)
  • \(\frac{d}{dx}(\cot x) = -\csc^2 x\)
  • \(\frac{d}{dx}(\sec x) = \sec x \tan x\)
  • \(\frac{d}{dx}(\csc x) = -\csc x \cot x\)

3. Inverse Trigonometric Derivatives

Inverse trigonometric derivatives appear frequently in modern NTA shift papers, often combined with chain rule transformations:

• \(\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}\)
• \(\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}\)

4. NTA Pattern and PYQ Weightage Analysis

Analysis of recent Previous Years Questions (PYQs) for JEE Mains reveals key trends in Calculus:

• Chapter Weightage: Calculus accounts for 8 to 10 questions per 30-question paper in JEE Mains. Limits, Continuity, and Differentiability alone consistently contribute 2 to 3 direct questions.
• NTA Pattern Insights: Recent shift papers focus on combining standard limit evaluation with Great Integer Functions (GIF) \(\lfloor x \rfloor\) and Fractional Part Functions \(\{x\}\). Additionally, L'Hôpital's Rule combined with Leibniz Rule for integration under the integral sign is a recurring advanced topic.
• Practice Recommendation: Solving over 1,000+ targeted PYQs builds the speed and accuracy necessary to solve calculus problems within 2 to 3 minutes during the actual exam.

5. Frequently Asked Questions (People Also Ask)