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CodeVID-M12-05-IMP-01
Implicit, Logarithmic & Parametric Differentiation — Assignment
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.
For $x^2+y^2=25$, $\dfrac{dy}{dx}=$
- A.$-\tfrac{x}{y}$
- B.$\tfrac{x}{y}$
- C.$-\tfrac{y}{x}$
- D.$\tfrac{y}{x}$
2.
$\dfrac{d}{dx}x^{x}=$
- A.$x\cdot x^{x-1}$
- B.$x^{x}(1+\ln x)$
- C.$x^{x}\ln x$
- D.$x^x$
3.
If $x=at^2,\ y=2at$, then $\dfrac{dy}{dx}=$
- A.$t$
- B.$\tfrac1t$
- C.$\tfrac{1}{2t}$
- D.$2t$
4.
$\dfrac{d^2}{dx^2}(x^3)=$
- A.$3x^2$
- B.$6x$
- C.$6$
- D.$x^2$
5.
In implicit differentiation, $y$ is treated as:
- A.a constant
- B.a function of $x$
- C.zero
- D.$x$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find $\dfrac{dy}{dx}$ if $xy=1$.
7.
Differentiate $y=x^{x}$.
8.
If $x=t^2,\ y=t^3$, find $\dfrac{dy}{dx}$.
9.
Find $\dfrac{d^2y}{dx^2}$ if $y=x^4$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find $\dfrac{dy}{dx}$ if $x^2+xy+y^2=0$.
11.
Differentiate $y=x^{\sin x}$.
12.
If $x=a\cos\theta,\ y=a\sin\theta$, find $\dfrac{dy}{dx}$.
13.
Find $\dfrac{dy}{dx}$ if $y=x^{\ln x}$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
If $x=a(\theta-\sin\theta),\ y=a(1-\cos\theta)$, find $\dfrac{dy}{dx}$.
15.
If $x^{y}=e^{x-y}$, prove that $\dfrac{dy}{dx}=\dfrac{\ln x}{(1+\ln x)^2}$.
Answer Key
Section A — Multiple Choice Questions
- (A) $-\tfrac{x}{y}$
- (B) $x^{x}(1+\ln x)$
- (B) $\tfrac1t$
- (B) $6x$
- (B) a function of $x$
Section B — Short Answer (2 marks)
- $-\dfrac{y}{x}$ (i.e. $-\tfrac{1}{x^2}$).
- $x^{x}(1+\ln x)$.
- $\dfrac{3t}{2}$.
- $12x^2$.
Section C — Short Answer (3 marks)
- $-\dfrac{2x+y}{x+2y}$.
- $x^{\sin x}\!\left(\cos x\ln x+\dfrac{\sin x}{x}\right)$.
- $-\cot\theta$.
- $x^{\ln x}\cdot\dfrac{2\ln x}{x}$.
Section D — Long Answer (5 marks)
- $\cot\dfrac{\theta}{2}$.
- $\dfrac{dy}{dx}=\dfrac{\ln x}{(1+\ln x)^2}$.
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