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 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
Core ConceptRead this first — the foundation of the topic
Core Concept
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
Exam PatternsWhat 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.
Worked ExampleSolve 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)
ShortcutsUse 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.
Exam TrapsCommon 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