NCERT Solutions for Class 12 Maths Exercise Miscellenous Chapter 5 Continuity and Differentiability – FREE PDF Download
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NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exercise Miscellenous (Ex Misc.)
Differentiate with respect to the functions in Exercises 1 to 5.
1.

=
2. 


=
3. 

4. 


=
=
=
5. 

dydx=−2x+7√.24−x2√.12−12.cos−1x22x+7√×2(2x+7)dydx=−2x+7.24−x2.12−12.cos−1x22x+7×2(2x+7)
=
=
= −[14−x2√2x+7√+cos−1x2(2x+7)32]−
Differentiate with respect to
the functions in Exercises 6 to 11.
6.

Now,
= =
And
= =
From eq. (i),
=
=
7. 

Taking log both sides, we get
=
=>1ydydx=logx1logx.1x+log(logx)1x=>1ydydx=logx1logx.1x+log(logx)1x
=>1ydydx=1x+log(logx)1x=>1ydydx=1x+log(logx)1x
=>1ydydx=[1+log(logx)x]=>1ydydx=
=
8.
for some constants
and 



dydx=(asinx−bcosx)sin(acosx+bsinx)dydx=(asinx−bcosx)sin(acosx+bsinx)
9. 

Taking log Both Sides, we get
=>logy=(sinx−cosx)log(sinx−cosx)=>logy=(sinx−cosx)log(sinx−cosx)
=> ddxlogy=(sinx−cosx)ddxlog(sinx−cosx)+log(sinx−cosx)ddx(sinx−cosx)ddxlogy=(sinx−cosx)ddxlog(sinx−cosx)+log(sinx−cosx)ddx(sinx−cosx)
10.
for some fixed
and 

= …….(i)
Now taking , let
……….(ii)
Taking log both sides, we get
=
=
From eq. (ii),
From eq. (i),
=> dydx=xx(1+logx)+axa−1+axlogadydx=xx(1+logx)+axa−1+axloga
11.
for 


Putting and
……….(i)
Now
=
=
……….(ii)
Again
=
=
……….(iii)
Putting the values from eq. (ii) and (iii) in eq. (i),
12. Find
if
and 


and dxdt=10ddt(t−sint)dxdt=10ddt(t−sint)
=>dxdt=10(1−cost)=>dxdt=10(1−cost)
=
=
=
13. Find
if 

=
=
= 0
14. If
for
prove that 

Squaring both sides,We get
dydx=(1+x)ddx(−x)−(−x)ddx(1+x)(1+x)2dydx=(1+x)ddx(−x)−(−x)ddx(1+x)(1+x)2
=> dydx=−(1+x)+x(1+x)2dydx=−(1+x)+x(1+x)2
= Proved.
15. If
for some
prove that
is a constant independent of a and b.

……….(ii)
Again
[From eq. (ii)
=
= ……….(iii)
Putting values of and
in the given expression,
=
= =
=
which is a constant and is independent of and
16. If
with
prove that 

=
=
17. If
and
find 


Differentiating both sides with respect to
and
and
and
and
Now
Again =
= =
=
18. If
show that
exists for all real
and find it.


Now, L.H.D. at
=
=
L.H.D. at
= R.H.D. at
Therefore, is differentiable at
.
Now, L.H.D. at
=
=
And R.H.D. at
=
limx→0−−3x=0limx→0−−3x=0
Again L.H.D. at
= R.H.D. at
19. Using mathematical induction, prove that
for all positive integers 

…..(i)
=
,
which is true as
Now we suppose is true.
…….(ii)
To establish the truth of we prove,
=
=
Therefore, is true if
is true but
is true.
By Principal of Induction
is true for all
N.
20. Using the fact that
and the differentiation, obtain the sum formula for cosines.

Consider A and B as function of and differentiating both sides w.r.t.
21. Does there exist a function which is continuous everywhere but not differentiable at exactly two points?

is continuous everywhere but it is not differentiable at
and
22. If
prove that 

=
23. If
show that 

=
=
Differentiating both sides with respect to
Proved.