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CDS Differentiation

Study Material — 1 PYQs (2019–2019) · Concept Notes · Shortcuts

CDS Differentiation is a frequently tested subtopic — 1 previous year questions from 2019–2019 papers are included below with concept notes, key rules and shortcut tricks.

1 PYQs

2019–2019

33 Practice

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10 Key Points

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Previous Year Questions

CDS Differentiation — Past Exam Questions

1 questions from actual CDS papers · all shown free · click option to reveal solution

Exam Q 12019Previous Year Pattern

If y = x^x, then dy/dx is equal to:

Concept Notes

Differentiation— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

If y = f(x), then dy/dx represents the derivative of y with respect to x. This derivative gives us the slope of the tangent line at any point on the curve

Power Rule

d/dx(x^n) = n × x^(n-1) 2

Constant Rule

d/dx(c) = 0 (where c is constant) 3

Sum Rule

d/dx(u + v) = du/dx + dv/dx 4

Product Rule

d/dx(uv) = u(dv/dx) + v(du/dx) 5

Quotient Rule

d/dx(u/v) = [v(du/dx) - u(dv/dx)]/v² 6

Chain Rule

d/dx[f(g(x))] = f'(g(x)) × g'(x) Standard Derivatives (Must Memorize): - d/dx(sin x) = cos x - d/dx(cos x) = -sin x - d/dx(tan x) = sec²x - d/dx(e^x) = e^x - d/dx(ln x) = 1/x - d/dx(a^x) = a^x × ln a

2.Exam Patterns

What examiners ask — read before attempting PYQs

NDA typically asks differentiation in three ways: 1. Direct differentiation of polynomials, trigonometric, and exponential functions 2. Composite functions requiring chain rule 3. Word problems involving rates of change 4.

Finding equation of tangent/normal at given points Shortcut for Chain Rule: Remember the 'outside-inside' method. Differentiate the outer function first, keep the inner function unchanged, then multiply by the derivative of the inner function.

3.Worked Example

Solve this step-by-step before moving on

1

Step 1

Apply power rule to each term

2

Step 2

d/dx(2x³) = 2 × 3x² = 6x²

3

Step 3

d/dx(5x²) = 5 × 2x = 10x

4

Step 4

d/dx(-3x) = -3

5

Step 5

d/dx(7) = 0 (constant)

6

Step 6

dy/dx = 6x² + 10x - 3 Worked Example 2: Find dy/dx if y = sin(3x² + 2)

1

Step 1

This is a composite function, use chain rule

2

Step 2

Outer function = sin u, where u = 3x² + 2

3

Step 3

d/dx(sin u) = cos u

4

Step 4

du/dx = d/dx(3x² + 2) = 6x

5

Step 5

dy/dx = cos(3x² + 2) × 6x = 6x cos(3x² + 2)

4.Shortcuts

Use these to save 30–60 seconds per question

for Product Rule: Remember 'FIRST × second' + 'first × SECOND'. This helps avoid confusion about which function to differentiate first. Another Speed Trick: For simple powers like (ax + b)^n, use the formula: derivative = n(ax + b)^(n-1) × a. This saves time compared to full chain rule expansion.

5.Exam Traps

Common mistakes students make — avoid these

Students forget to apply the chain rule when dealing with composite functions. For example, when differentiating sin(2x), many write cos(2x) instead of 2cos(2x). Always check if there's an 'inner function' and multiply by its derivative. Another frequent error is sign mistakes in trigonometric derivatives.

Remember that d/dx(cos x) = -sin x (negative sign), not +sin x. For NDA success, practice mixed problems daily. Focus on recognizing patterns quickly - whether to use power rule, product rule, or chain rule. Speed and accuracy in basic derivatives will save precious exam time for complex application problems.

Key Points to Remember

Power Rule: d/dx(x^n) = n × x^(n-1) - most frequently used formula
Chain Rule shortcut: differentiate outside function, keep inside unchanged, multiply by inside derivative
Product Rule memory trick: FIRST × second' + first' × SECOND
Derivative of sin x = cos x, derivative of cos x = -sin x (note the negative)
Derivative of any constant is always zero
For (ax + b)^n: derivative = n(ax + b)^(n-1) × a
Sum rule allows differentiating each term separately
Derivative of e^x is e^x itself (unique property)
Quotient rule: top×bottom' minus bottom×top', all over bottom squared
Always check for composite functions to avoid missing chain rule application

Exam-Specific Tips

d/dx(tan x) = sec²x
d/dx(ln x) = 1/x
d/dx(e^x) = e^x
d/dx(a^x) = a^x × ln a
Quotient rule formula: d/dx(u/v) = [v(du/dx) - u(dv/dx)]/v²
Chain rule: d/dx[f(g(x))] = f'(g(x)) × g'(x)
d/dx(cot x) = -cosec²x
d/dx(sec x) = sec x tan x

Practice MCQs

Differentiation — Practice Questions

33graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 33

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Practice 1easy

If y = (3x² + 2x + 1)⁵, then dy/dx at x = 1 is equal to:

Practice 2easy

If y = x·sin(x), then d²y/dx² is equal to:

Practice 3easy

If y = ln(x² + 3x + 2), then dy/dx is equal to:

Practice 4easy

If x = t³ - 3t and y = t² + 2t, then dy/dx at t = 1 is equal to:

Practice 5easy

If y = e^(2x)·cos(3x), then dy/dx at x = 0 is equal to:

Practice 6easy

If x = t² + 2t and y = t³ - 1, then dy/dx at t = 1 is equal to:

Practice 7easy

If y = e^(2x)·cos(3x), then dy/dx is equal to:

Practice 8easy

If y = √(x³ + 4x), then dy/dx is equal to:

Practice 9easy

If y = ln(x² + 1), then dy/dx at x = 1 is equal to:

Practice 10medium

If x = 2t³ and y = 3t² - 1, then dy/dx at t = 1 is:

Practice 11medium

If y = log(sin(x)), then dy/dx is equal to:

Practice 12medium

If y = (e^x)/(x + 1), then dy/dx is equal to:

Practice 13medium

If x = 2t + 3 and y = t² - 1, then dy/dx at t = 2 is:

Practice 14medium

If y = e^(x²), then d/dx[e^(x²)] is equal to:

Practice 15medium

If y = (sin x)/(1 + cos x), then dy/dx is equal to:

Practice 16medium

If x = t² + 2t and y = t³ - 1, then dy/dx at t = 1 is:

Practice 17medium

If y = (x² + 3x + 1)⁵, then dy/dx at x = 1 is equal to:

Practice 18medium

If y = x·sin(x), then the second derivative d²y/dx² is:

Practice 19hard

Let f(x) = x·e^(-x²). Find the x-coordinate of the point where f(x) has a local maximum.

Practice 20hard

Let f(x) = (x² + 1)⁵ · sin(x). Find f'(x) and evaluate f'(0).

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60-Second Revision — Differentiation

Remember: Always check for composite functions and apply chain rule
Formula: Power rule d/dx(x^n) = n × x^(n-1) covers 60% of basic problems
Trap: Don't forget negative sign in d/dx(cos x) = -sin x
Speed trick: For (ax + b)^n, derivative is n(ax + b)^(n-1) × a
Product rule: FIRST × second' + first' × SECOND
Remember: Derivative of constant is zero, derivative of e^x is e^x
Check: After solving, verify if chain rule was needed for composite functions

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