ilovepdf_merged (6).pdf). Content covers sections 27.1–27.x.Derivatives of sin, cos, tan and related functions, combined with the chain rule for compositions like sin(ax+b).
Differentiate products like x sin x using product rule; quotients like tan x / x using quotient rule. Simplify using identities only after differentiating if it helps.
Most exam-important points from this chapter:
Start every answer with the key definition or standard form from Differentiation of Trigonometric Functions.
Copy the formula, then substitute values; never jump to the number alone.
Check domain, quadrant, non-zero denominators, and applicability of the theorem.
Memorise and apply: sin 2
Memorise and apply: lim cos
English questions from NIOS Mathematics (311) public / sample papers (Sample 2024, Apr 2024, Oct 2024, Apr 2025; board-style where scanned papers had no extractable text). Mapped exclusively to this chapter — no cross-chapter duplicates. Use Model Answer for the marking key and Explanation for working.
4 question(s) · Sources: Board-style, Board-style chain, Board-style product, Sample QP 2024
PYQ1. If sin y = x sin(a+y), find dy/dx (in terms of a and y).
Model Answer
1) dy/dx = sin a / sin²(a+y)
Explanation
1) Differentiate: cos y · y′ = sin(a+y) + x cos(a+y)·y′. Also x=sin y / sin(a+y). Standard result: y′ = sin a / sin²(a+y).
PYQ2. Differentiate y = sin x + cos x.
Model Answer
1) cos x − sin x
Explanation
1) (sin)′=cos, (cos)′=−sin.
PYQ3. Find d/dx tan(3x).
Model Answer
1) 3 sec²(3x)
Explanation
1) d/dx tan u = sec² u · u′ with u=3x ⇒ 3 sec²(3x).
PYQ4. Differentiate y = x cos x.
Model Answer
1) cos x − x sin x
Explanation
1) Product rule: 1·cos x + x(−sin x).
10 English exam-style problems for this chapter (NIOS Mathematics 311). Each question states the formula, gives full step-by-step working with the final answer, plus formal/easy explanations. English only.
Differentiate y = sin x + cos x.
1) y′ = cos x − sin x.
Answer: cos x − sin x
Standard derivatives of sine and cosine.
cos stays-ish for sin; cos becomes −sin.
Radians assumed.
Sign on −sin x.
Find d/dx tan(3x).
1) Chain rule: sec²(3x)·3 = 3 sec²(3x).
Answer: 3 sec²(3x)
Derivative of tan u is sec² u · u′.
sec² of inside times 3.
Defined where cos(3x)≠0.
Do not omit the 3.
Differentiate y = x cos x.
1) y′ = cos x + x(−sin x) = cos x − x sin x.
Answer: cos x − x sin x
Product of x and cos x.
1·cos + x·(−sin).
Can write as cos x − x sin x.
Second term negative.
d/dx sin(2x+π/4).
1) 2 cos(2x+π/4).
Answer: 2 cos(2x + π/4)
Chain rule on sine of a linear expression.
Cos of the same angle times 2.
Keep the phase π/4.
Factor is 2 not 1.
Differentiate sec x.
1) d/dx sec x = sec x tan x.
Answer: sec x tan x
Standard derivative of secant.
sec times tan.
Equivalently write 1/cos and quotient rule.
Not sec² x (that is tan).
If y=sin x, find dy/dx at x=π/3.
1) y′=cos x; cos(π/3)=1/2.
Answer: 1/2
Evaluate the derivative function at the given point.
cos of 60° is 1/2.
Do not evaluate sin at π/3 for the derivative value.
π/3 not 3.
Differentiate y = sin 4x.
1) y′ = 4 cos 4x.
Answer: 4 cos 4x
Chain rule with argument 4x.
cos of same angle times 4.
Radians assumed.
Check each algebraic step carefully.
Differentiate y = tan x − x.
1) y′ = sec² x − 1.
Answer: sec² x − 1
Derivative of tan minus 1.
sec² x − 1 = tan² x (optional identity).
Valid where cos x ≠ 0.
Check each algebraic step carefully.
Differentiate y = x sin x.
1) y′ = sin x + x cos x.
Answer: sin x + x cos x
Product of x and sin x.
1·sin + x·cos.
Do not treat as chain only.
Check each algebraic step carefully.
If y = cos x, find dy/dx at x = π/2.
1) y′ = −sin x; −sin(π/2) = −1.
Answer: −1
Differentiate then substitute.
cos becomes −sin; sin(π/2)=1.
Order: derivative first.
Check each algebraic step carefully.