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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-05-DIF-01
Differentiability & Chain Rule — Assignment
Chapter: Continuity and Differentiability
Topic: Differentiability & the Chain Rule
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
$\dfrac{d}{dx}\sin(x^2)=$
  • A.$\cos(x^2)$
  • B.$2x\cos(x^2)$
  • C.$2x\sin(x^2)$
  • D.$\cos(2x)$
2.
If $f$ is differentiable at $c$, then $f$ is:
  • A.discontinuous
  • B.continuous at $c$
  • C.constant
  • D.zero at $c$
3.
$\dfrac{d}{dx}(x^2e^x)=$
  • A.$2xe^x$
  • B.$e^x(x^2+2x)$
  • C.$x^2e^x$
  • D.$2xe^x+x$
4.
$f(x)=|x|$ at $x=0$ is:
  • A.differentiable
  • B.continuous but not differentiable
  • C.discontinuous
  • D.neither
5.
$\dfrac{d}{dx}\tan x=$
  • A.$\sec^2x$
  • B.$\sec x\tan x$
  • C.$-\csc^2x$
  • D.$\cos^2x$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Differentiate $(2x+1)^3$.
7.
Differentiate $\sin(3x)$.
8.
Differentiate $e^{5x}$.
9.
Differentiate $\ln(x^2)$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Differentiate $(x^2+1)^5$.
11.
Differentiate $x\sin x$.
12.
Differentiate $\dfrac{\sin x}{x}$.
13.
Show that $f(x)=|x|$ is not differentiable at $x=0$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
If $y=\sin(\sqrt{x})$, find $\dfrac{dy}{dx}$.
15.
Differentiate $y=e^{x}\cos x$.

Answer Key

Section A — Multiple Choice Questions
  1. (B) $2x\cos(x^2)$
  2. (B) continuous at $c$
  3. (B) $e^x(x^2+2x)$
  4. (B) continuous but not differentiable
  5. (A) $\sec^2x$
Section B — Short Answer (2 marks)
  1. $6(2x+1)^2$.
  2. $3\cos(3x)$.
  3. $5e^{5x}$.
  4. $\dfrac{2}{x}$.
Section C — Short Answer (3 marks)
  1. $10x(x^2+1)^4$.
  2. $\sin x+x\cos x$.
  3. $\dfrac{x\cos x-\sin x}{x^2}$.
  4. LHD $=-1$, RHD $=+1$; not differentiable.
Section D — Long Answer (5 marks)
  1. $\dfrac{\cos\sqrt{x}}{2\sqrt{x}}$.
  2. $e^{x}(\cos x-\sin x)$.
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