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Differential Calculus

Mathematics

Useful for: UPSC NDA/NA

Overview

Differential calculus studies rates of change using derivatives, with applications to slopes, maxima and minima.

Key points

d/dx (xⁿ) = n xⁿ⁻¹.
d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x.
Product rule: (uv)′ = u′v + uv′; quotient rule: (u/v)′ = (u′v − uv′)/v².
Chain rule: d/dx f(g(x)) = f′(g(x)) · g′(x).
At a maximum f′ = 0 and f″ < 0; at a minimum f′ = 0 and f″ > 0.
The derivative gives the slope of the tangent.

Practice MCQs

From the perspective of definition and scope, which one of the following statements about Differential Calculus is correct?
1. Differential Calculus is unrelated to institutions, processes or real-world applications.
2. Differential Calculus has no exceptions or limitations in exam analysis.
3. Differential Calculus questions reward a formula-first approach and accurate algebraic manipulation.
4. Do not apply a memorized Differential Calculus formula without checking its conditions.
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Answer: (c) Differential Calculus questions reward a formula-first approach and accurate algebraic manipulation.. Differential Calculus questions reward a formula-first approach and accurate algebraic manipulation.
Consider the following statements on Differential Calculus, focusing on definition and scope: 1. Do not apply a memorized Differential Calculus formula without checking its conditions. 2. Differential Calculus questions reward a formula-first approach and accurate algebraic manipulation. Which of the statements given above is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
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Answer: (b) 2 only. Statement 2 is correct; Statement 1 is a common misconception.

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Notes and questions are written by Kundali IQ for practice. They are not official papers; always check the latest official notification for the syllabus and pattern.