Calculus Cheat Sheet - NDA Mathematics Study Material & Formulas

1. Limits, Continuity & Differentiability

Limits form the foundational basis of differential calculus. Questions in NDA frequently evaluate indeterminate forms using fundamental definitions or specialized shortcuts.

Definition of Derivative

The derivative of a function \(f(x)\) at any point is defined using the first principle limit:

L'Hôpital's Rule

When evaluating limits that result in indeterminate forms such as \(\frac{0}{0}\) or \(\frac{\pm \infty}{\pm \infty}\), differentiate the numerator and denominator independently:

2. Differential Calculus & Derivative Rules

Standard algebraic differentiation rules allow quick calculation of derivatives without resorting to first principles.

Core Rules of Differentiation

• Power Rule: \(\frac{d}{dx}[x^n] = n x^{n-1}\)
• Product Rule: \((fg)' = f'g + fg'\)
• Quotient Rule: \((\frac{f}{g})' = \frac{f'g - fg'}{g^2}\)
• Chain Rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)

3. Trigonometric Differentiation

Trigonometric derivatives appear routinely in NDA problems involving composite functions, implicit differentiation, and limits.

Standard Derivatives

• \(\frac{d}{dx}[\sin x] = \cos x\)
• \(\frac{d}{dx}[\cos x] = -\sin x\)
• \(\frac{d}{dx}[\tan x] = \sec^2 x\)
• \(\frac{d}{dx}[\csc x] = -\csc x \cot x\)
• \(\frac{d}{dx}[\sec x] = \sec x \tan x\)
• \(\frac{d}{dx}[\cot x] = -\csc^2 x\)

4. Integration Techniques

Integral calculus problems in NDA often demand strategic substitution or algebraic simplification prior to integrating.

Integration by Parts

Used for integrating the product of two functions according to the ILATE rule hierarchy:

Substitution Method

Simplifies integrals by transforming the variable of integration:

5. Standard & Indefinite Integrals

Memorizing key antiderivatives is critical for speed during the 150-minute NDA Mathematics exam.

Fundamental Formula List

• \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n eq -1)\)
• \(\int \frac{1}{x} \, dx = \ln|x| + C\)
• \(\int e^x \, dx = e^x + C\)
• \(\int \sin x \, dx = -\cos x + C\)
• \(\int \cos x \, dx = \sin x + C\)
• \(\int \sec^2 x \, dx = \tan x + C\)

6. Infinite Series & Taylor Expansions

Infinite series and power series expansions help simplify complex limit evaluations and approximation problems in NDA calculus questions.

Geometric Series

Sum of an infinite geometric progression when the absolute common ratio is less than 1:

Taylor & Maclaurin Series

The general Taylor series expansion of a smooth function \(f(x)\) around \(x = a\):

Common Maclaurin expansions (centered at \(a = 0\)):

• \(e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \dots\)
• \(\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}\)
• \(\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\)

7. NDA Exam Pattern & Weightage Analysis

Understanding exam-specific patterns enables high-yield preparation for the NDA Mathematics paper.

• Chapter Weightage: Differential Calculus accounts for roughly 12-15 questions, while Integral Calculus typically contributes 10-12 questions.
• PYQ Trends: L'Hôpital's rule, simple definite integral properties, and standard derivative applications are tested almost every year.
• Time Strategy: Direct formula-based questions should take no more than 30-45 seconds each. Utilizing cheat sheet formulas minimizes steps significantly.

8. Frequently Asked Questions (People Also Ask)