Chapter Quizzes

NDA Calculus Master Guide: Limits, Derivatives, Integrals & Differential Equations

By SenaPrep Mathematics Faculty October 08, 2026 16 min read

Calculus represents the largest, highest-scoring domain in the UPSC NDA Mathematics syllabus. Spanning Differential Calculus, Integral Calculus, and Differential Equations, this block generates 25 to 35 questions (62.5 to 87.5 Marks) in Paper 1. While many aspirants find Calculus intimidating, UPSC questions in this section follow rigid, repeatable patterns that can be solved swiftly using standard theorems and shortcut rules. Compare this with our NDA Mathematics Weightage Breakdown.

The Calculus Rank Decider

Unlike Algebra where questions can be lengthy, Calculus questions in NDA test conceptual agility. Mastering L'Hôpital's Rule for Limits, the Chain Rule for Derivatives, King's Property for Definite Integrals, and Integrating Factors for Differential Equations will help you solve 80% of Calculus problems in under 60 seconds each.

1. Master Calculus Formula & Theorems Cheatsheet

A. Standard Limits & Indeterminate Forms

$\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{\tan x}{x} = 1, \quad \lim_{x \to 0} \frac{\sin^{-1}x}{x} = 1, \quad \lim_{x \to 0} \frac{\tan^{-1}x}{x} = 1$ — Practice Limits Questions
$\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}, \quad \lim_{x \to 0} \frac{1 - \cos(kx)}{x^2} = \frac{k^2}{2}$
$\lim_{x \to 0} \frac{e^x - 1}{x} = 1, \quad \lim_{x \to 0} \frac{a^x - 1}{x} = \ln a, \quad \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$
$\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}$
$1^\infty$ Exponential Limit Shortcut: If $\lim_{x \to a} f(x) = 1$ and $\lim_{x \to a} g(x) = \infty$, then $\lim_{x \to a} [f(x)]^{g(x)} = e^{\lim_{x \to a} g(x)[f(x)-1]}$.
L'Hôpital's Rule: If $\lim_{x \to c} \frac{f(x)}{g(x)}$ yields $\frac{0}{0}$ or $\frac{\pm\infty}{\pm\infty}$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$, repeatedly until the indeterminate form resolves.

B. Standard Derivatives & Rules of Differentiation

$\frac{d}{dx}(x^n) = n x^{n-1}, \quad \frac{d}{dx}(e^x) = e^x, \quad \frac{d}{dx}(a^x) = a^x \ln a, \quad \frac{d}{dx}(\ln x) = \frac{1}{x}$ — Practice Derivatives MCQs
$\frac{d}{dx}(\sin x) = \cos x, \quad \frac{d}{dx}(\cos x) = -\sin x, \quad \frac{d}{dx}(\tan x) = \sec^2 x, \quad \frac{d}{dx}(\cot x) = -\csc^2 x$
$\frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}(\tan^{-1}x) = \frac{1}{1+x^2}, \quad \frac{d}{dx}(\sec^{-1}x) = \frac{1}{|x|\sqrt{x^2-1}}$
Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$
Product & Quotient Rule: $\frac{d}{dx}(uv) = u'v + uv', \quad \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$
Logarithmic Differentiation: For $y = [u(x)]^{v(x)} \implies \frac{dy}{dx} = y \left[ v'\ln u + \frac{v u'}{u} \right]$.

C. Applications of Derivatives: Maxima, Minima & Tangents

Slope of Tangent: $m = \left.\frac{dy}{dx}\right|_{(x_1, y_1)}$. Equation of Tangent: $y - y_1 = m(x - x_1)$.
Slope of Normal: $m_N = -\frac{1}{m}$. Equation of Normal: $y - y_1 = -\frac{1}{m}(x - x_1)$.
Increasing / Decreasing: $f(x)$ is strictly increasing if $f'(x) > 0$, and strictly decreasing if $f'(x) < 0$ — Practice Increasing/Decreasing MCQs.
Local Maxima / Minima: Find critical points by setting $f'(x) = 0$. If $f''(c) < 0$, $x = c$ is a Local Maximum; if $f''(c) > 0$, $x = c$ is a Local Minimum — Practice Maxima & Minima MCQs.

D. Standard Integrals & Definite Integral Properties

$\int x^n dx = \frac{x^{n+1}}{n+1} + C \ (n \neq -1), \quad \int \frac{1}{x} dx = \ln|x| + C, \quad \int e^x dx = e^x + C$ — Practice Indefinite Integration
$\int \frac{dx}{x^2 + a^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C, \quad \int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\left(\frac{x}{a}\right) + C$
$\int e^x [f(x) + f'(x)] dx = e^x f(x) + C$
King's Property: $\int_a^b f(x) dx = \int_a^b f(a + b - x) dx \implies \int_0^a f(x) dx = \int_0^a f(a - x) dx$ — Practice Definite Integration.
Odd / Even Property: $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$ (if $f(-x) = f(x)$) and $= 0$ (if $f(-x) = -f(x)$).
Area Under Curve: Area between $y^2 = 4ax$ and $y = mx$ is $\frac{8a^2}{3m^3}$. Area between $y^2 = 4ax$ and $x^2 = 4by$ is $\frac{16ab}{3}$ — Practice Area Under Curves.

E. Differential Equations

Order & Degree: Order = Highest derivative. Degree = Power of the highest derivative (when polynomial in derivatives) — Practice Order & Degree MCQs.
Linear First Order D.E.: $\frac{dy}{dx} + P(x)y = Q(x) \implies \text{Integrating Factor } IF = e^{\int P(x) dx}$ — Practice First Order D.E..
General Solution: $y \cdot IF = \int (Q(x) \cdot IF) dx + C$.

2. High-Speed Calculus Shortcuts for NDA

Hack 1: The King's Property Direct $\pi/4$ Result

For any integral of the form: $$\int_0^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} dx = \int_0^{\pi/2} \frac{\tan^n x}{1 + \tan^n x} dx = \int_0^{\pi/2} \frac{1}{1 + \cot^n x} dx = \frac{\pi}{4}$$ The answer is always half the upper limit: $\frac{\pi/2 - 0}{2} = \frac{\pi}{4}$, regardless of the exponent $n$. Try this on our Solved NDA PYQs Archive.

Hack 2: Instant $e^x[f(x)+f'(x)]$ Recognition

Whenever you see $e^x$ multiplied by an algebraic fraction or trigonometric sum inside an integral, differentiate the first term mentally. If the second term matches its derivative, the integral is directly $e^x \cdot \text{first term} + C$.
Example: $\int e^x [\tan x + \sec^2 x] dx = e^x\tan x + C$.

Interactive Practice Quiz

Test your preparation live! Choose an option for each question to see instant feedback and deep step-by-step explanations.

Question 1 of 10 Live App Practice

What is ∫₀¹ ln(1/x − 1) dx equal to?

Question 2 of 10 Live App Practice

What is the range of the function f(x) = l - sin x defined on entire real line?

Question 3 of 10 Live App Practice

Consider the following statements in respect of the function

f(x) = sin(1/x²), x ≠ 0 :

1. It is continuous at x = 0, if f(0) = 0.
2. It is continuous at x = 2/√π.

Which of the above statements is/are correct?

Question 4 of 10 Live App Practice

If f(x) = [x]/|x|, x ≠ 0 where [ ] denotes the greatest integer function, then what is the right-hand limit of f(x) at x = 1?

Question 5 of 10 Live App Practice

Consider the following statements in respect of
f(x) = |x| − 1 :

1. f(x) is continuous at x = 1.
2. f(x) is differentiable at x = 0.

Which of the above statements is/are correct?

Question 6 of 10 Live App Practice

Question 7 of 10 Live App Practice

What is lim(n→∞) (aⁿ + bⁿ)/(aⁿ − bⁿ) where a > b > 1, equal to?

Question 8 of 10 Live App Practice

Let f(x) be a polynomial function such that f(x) = x⁴. What is f′(1) equal to?

Question 9 of 10 Live App Practice

Question 10 of 10 Live App Practice

If y = (1 + x)(1 + x²)(1 + x⁴)(1 + x⁸)(1 + x¹⁶), then what is dy/dx at x = 0 equal to?

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Frequently Asked Questions

Which is more important for NDA: Differential Calculus or Integral Calculus?

Both carry almost equal weight. Differential Calculus (Limits, Continuity, Derivatives, Applications) accounts for ~15 questions, and Integral Calculus with Differential Equations accounts for ~15 questions. Because integration directly builds upon differentiation, mastering both sequentially is vital.

How can I solve NDA limits problems faster?

Memorize standard limit identities like (1 - cos kx)/x^2 = k^2 / 2 and apply L'Hôpital's Rule for 0/0 or infinity/infinity forms. For 1^infinity forms, apply the shortcut e^[limit g(x)(f(x)-1)].

How many questions are asked from Differential Equations in NDA?

UPSC typically asks 4 to 6 questions on Differential Equations every year, focusing primarily on finding the Order and Degree, solving first-order linear differential equations with Integrating Factors, and variable separable forms.