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.

• Continuity and Differentiability Class 12 Ex 5.1
• Continuity and Differentiability Class 12 Ex 5.2
• Continuity and Differentiability Class 12 Ex 5.3
• Continuity and Differentiability Class 12 Ex 5.4
• Continuity and Differentiability Class 12 Ex 5.5
• Continuity and Differentiability Class 12 Ex 5.6
• Continuity and Differentiability Class 12 Ex 5.8

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.

Ex 5.7 Class 12 Maths Question 1.
x² + 3x + 2 = y(say)
Solution:
\(\frac { dy }{ dx } =2x+3\quad and\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =2\)

Ex 5.7 Class 12 Maths Question 2.
x²⁰ = y(say)
Solution:
\(\frac { dy }{ dx } ={ 20 }x^{ 19 }\quad =>\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =20\times { 19x }^{ 18 }={ 380 }x^{ 18 }\qquad \)

Ex 5.7 Class 12 Maths Question 3.
x.cos x = y(say)
Solution:
\(\frac { dy }{ dx } =x(-sinx)+cosx.1,=-xsinx+cosx\)
\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =-xcosx-sinx-sinx=-xcosx-2sinx\)

Ex 5.7 Class 12 Maths Question 4.
log x = y (say)
Solution:
\(\frac { dy }{ dx } =\frac { 1 }{ x } =>\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =-\frac { 1 }{ { x }^{ 2 } } \)

Ex 5.7 Class 12 Maths Question 5.
x³ log x = y (say)
Solution:
x³ log x = y
\(=>\frac { dy }{ dx } ={ x }^{ 3 }.\frac { 1 }{ x } +logx\times { 3x }^{ 2 }={ x }^{ 2 }+{ 3x }^{ 2 }logx \)
\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =2x+{ 3x }^{ 2 }.\frac { 1 }{ x } +logx.6x=x(5+6logx) \)

Ex 5.7 Class 12 Maths Question 6.
e^x sin5x = y
Solution:
e^x sin5x = y

Ex 5.7 Class 12 Maths Question 7.
e^6x cos3x = y
Solution:
e^6x cos3x = y

Ex 5.7 Class 12 Maths Question 8.
tan^-1 x = y
Solution:
\(\frac { dy }{ dx } =\frac { 1 }{ 1+{ x }^{ 2 } } =>\frac { { d }^{ 2y } }{ { dx }^{ 2 } } =\frac { -2x }{ { ({ 1+x }^{ 2 }) }^{ 2 } } \)

Ex 5.7 Class 12 Maths Question 9.
log(logx) = y
Solution:
log(logx) = y
\(\frac { dy }{ dx } =\frac { 1 }{ logx } .\frac { 1 }{ x } \)

Ex 5.7 Class 12 Maths Question 10.
sin(log x) = y
Solution:
sin(log x) = y
\(\frac { dy }{ dx } =\frac { cos(logx) }{ x } \)
\(and\quad \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =\frac { x.\left[ -sin(logx) \right] .\frac { 1 }{ x } -cos(logx).1 }{ { x }^{ 2 } } \)
\(=\frac { \left[ sin(logx)+cos(logx) \right] }{ { x }^{ 2 } } \)

Ex 5.7 Class 12 Maths 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

Ex 5.7 Class 12 Maths Question 12.
If y = cos^-1 x, Find \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \) in terms of y alone.
Solution:
\(\frac { dy }{ dx } =-{ \left( { 1-x }^{ 2 } \right) }^{ -\frac { 1 }{ 2 } }\)
\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =\frac { -cosy }{ { \left( { sin }^{ 2 }y \right) }^{ \frac { 3 }{ 2 } } } =-coty\quad { cosec }^{ 2 }y\)

Ex 5.7 Class 12 Maths 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)

Ex 5.7 Class 12 Maths 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
\(\quad y=A{ e }^{ mx }+B{ e }^{ nx },\quad \)

Ex 5.7 Class 12 Maths 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

Ex 5.7 Class 12 Maths Question 16.
If e^y(x+1) = 1,show that \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ \left( \frac { dy }{ dx } \right) }^{ 2 }\)
Solution:
\({ e }^{ y }(x+1)=1=>{ e }^{ y }=\frac { 1 }{ x+1 } \)

Ex 5.7 Class 12 Maths 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

We hope the NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 help you. If you have any query regarding NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7, drop a comment below and we will get back to you at the earliest.