NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7.

Board	CBSE
Textbook	NCERT
Class	Class 12
Subject	Maths
Chapter	Chapter 5
Chapter Name	Continuity and Differentiability
Exercise	Ex 5.7
Number of Questions Solved	17
Category	NCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exc 5.7

Find the second order derivatives of the functions given in Questions 1 to 10.

Question 1.
x² + 3x + 2 = y(say)
Solution:
$\frac { dy }{ dx } =2x+3\quad and\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =2$

Question 2.
x²⁰ = y(say)
Solution:
$\frac { dy }{ dx } ={ 20 }x^{ 19 }\quad =>\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =20\times { 19x }^{ 18 }={ 380 }x^{ 18 }\qquad$

Question 3.
x.cos x = y(say)
Solution:
$\frac { dy }{ dx } =x(-sinx)+cosx.1,=-xsinx+cosx$
$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =-xcosx-sinx-sinx=-xcosx-2sinx$

Question 4.
log x = y (say)
Solution:
$\frac { dy }{ dx } =\frac { 1 }{ x } =>\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =-\frac { 1 }{ { x }^{ 2 } }$

Question 5.
x³ log x = y (say)
Solution:
x³ log x = y
$=>\frac { dy }{ dx } ={ x }^{ 3 }.\frac { 1 }{ x } +logx\times { 3x }^{ 2 }={ x }^{ 2 }+{ 3x }^{ 2 }logx$
$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =2x+{ 3x }^{ 2 }.\frac { 1 }{ x } +logx.6x=x(5+6logx)$

Question 6.
e^x sin5x = y
Solution:
e^x sin5x = y
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability 6

Question 7.
e^6x cos3x = y
Solution:
e^6x cos3x = y
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability 7

Question 8.
tan^-1 x = y
Solution:
$\frac { dy }{ dx } =\frac { 1 }{ 1+{ x }^{ 2 } } =>\frac { { d }^{ 2y } }{ { dx }^{ 2 } } =\frac { -2x }{ { ({ 1+x }^{ 2 }) }^{ 2 } }$

Question 9.
log(logx) = y
Solution:
log(logx) = y
$\frac { dy }{ dx } =\frac { 1 }{ logx } .\frac { 1 }{ x }$
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability 9

Question 10.
sin(log x) = y
Solution:
sin(log x) = y
$\frac { dy }{ dx } =\frac { cos(logx) }{ x }$
$and\quad \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =\frac { x.\left[ -sin(logx) \right] .\frac { 1 }{ x } -cos(logx).1 }{ { x }^{ 2 } }$
$=\frac { \left[ sin(logx)+cos(logx) \right] }{ { x }^{ 2 } }$

Question 11.
If y = 5 cosx – 3 sin x, prove that $\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +y=0$
Solution:
$\frac { dy }{ dx } =-5sinx-3cosx$
$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =-5cosx+3sinx=-y$
$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +y=0$
Hence proved

Question 12.
If y = cos^-1 x, Find $\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }$ in terms of y alone.
Solution:
$\frac { dy }{ dx } =-{ \left( { 1-x }^{ 2 } \right) }^{ -\frac { 1 }{ 2 } }$
$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =\frac { -cosy }{ { \left( { sin }^{ 2 }y \right) }^{ \frac { 3 }{ 2 } } } =-coty\quad { cosec }^{ 2 }y$

Question 13.
If y = 3 cos (log x) + 4 sin (log x), show that
${ x }^{ 2 }{ y }_{ 2 }+{ xy }_{ 1 }+y=0$
Solution:
Given that
y = 3 cos (log x) + 4 sin (log x)
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability 13

Question 14.
$If\quad y=A{ e }^{ mx }+B{ e }^{ nx },\quad show\quad that\quad \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -(m+n)\frac { dy }{ dx } +mny=0$
Solution:
Given that
$\quad y=A{ e }^{ mx }+B{ e }^{ nx },\quad$
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability 14

Question 15.
If y = 500e^7x + 600e^-7x, show that $\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =49y$ .
Solution:
we have
y = 500e^7x + 600e^-7x
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability 15

Question 16.
If e^y(x+1) = 1,show that $\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ \left( \frac { dy }{ dx } \right) }^{ 2 }$
Solution:
${ e }^{ y }(x+1)=1=>{ e }^{ y }=\frac { 1 }{ x+1 }$
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability 16

Question 17.
If y=(tan^-1 x)² show that (x²+1)²y₂+2x(x²+1)y₁=2
Solution:
we have
y=(tan^-1 x)²
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability 17

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NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exc 5.7

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