NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability Ex 5.5
Page No 178:
Question 1:
Differentiate the function with respect to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6914/Chapter%205_html_5195ccf8.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6914/Chapter%205_html_a90b370.gif)
Taking logarithm on both the sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6914/Chapter%205_html_m659895a.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6914/Chapter%205_html_4069ee03.gif)
Question 2:
Differentiate the function with respect to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6915/Chapter%205_html_630e908.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6915/Chapter%205_html_756604ee.gif)
Taking logarithm on both the sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6915/Chapter%205_html_m7d328c07.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6915/Chapter%205_html_m6c60117d.gif)
Question 3:
Differentiate the function with respect to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6918/Chapter%205_html_783d3a39.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6918/Chapter%205_html_51ab72c3.gif)
Taking logarithm on both the sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6918/Chapter%205_html_m7f8be8c7.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6918/Chapter%205_html_m37df01e.gif)
Question 4:
Differentiate the function with respect to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6922/Chapter%205_html_71de10.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6922/Chapter%205_html_m2a70b6e1.gif)
u = xx
Taking logarithm on both the sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6922/Chapter%205_html_4a39af9.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6922/Chapter%205_html_m278d76e2.gif)
v = 2sin x
Taking logarithm on both the sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6922/Chapter%205_html_md5917e4.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6922/Chapter%205_html_m363ba02e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6922/Chapter%205_html_63a5305b.gif)
Question 5:
Differentiate the function with respect to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6926/Chapter%205_html_m173da24a.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6926/Chapter%205_html_m23f8f68.gif)
Taking logarithm on both the sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6926/Chapter%205_html_4d1e6258.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6926/Chapter%205_html_2e65f58d.gif)
Question 6:
Differentiate the function with respect to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6929/Chapter%205_html_6db13fca.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6929/Chapter%205_html_5ce4fc8c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6929/Chapter%205_html_10dec314.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6929/Chapter%205_html_m67161841.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6929/Chapter%205_html_4d0680ca.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6929/Chapter%205_html_m291a6d50.gif)
Therefore, from (1), (2), and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6929/Chapter%205_html_1e132b57.gif)
Question 7:
Differentiate the function with respect to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6932/Chapter%205_html_m1b6d35e3.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6932/Chapter%205_html_m31ae62d8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6932/Chapter%205_html_m533a693d.gif)
u = (log x)x
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6932/Chapter%205_html_45b75a57.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6932/Chapter%205_html_m3bd5e52e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6932/Chapter%205_html_55f8f753.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6932/Chapter%205_html_m3e924d53.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6932/Chapter%205_html_5324cf14.gif)
Therefore, from (1), (2), and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6932/Chapter%205_html_m3d5fae4.gif)
Question 8:
Differentiate the function with respect to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6935/Chapter%205_html_1a8d3394.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6935/Chapter%205_html_1596bc50.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6935/Chapter%205_html_34ed9641.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6935/Chapter%205_html_m4cfb3914.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6935/Chapter%205_html_m220de476.gif)
Therefore, from (1), (2), and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6935/Chapter%205_html_66e67501.gif)
Question 9:
Differentiate the function with respect to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6937/Chapter%205_html_263eebdd.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6937/Chapter%205_html_731c8f29.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6937/Chapter%205_html_555a94f9.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6937/Chapter%205_html_me8cba15.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6937/Chapter%205_html_m4369494.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6937/Chapter%205_html_m61700c16.gif)
From (1), (2), and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6937/Chapter%205_html_54c0cb1e.gif)
Question 10:
Differentiate the function with respect to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6942/Chapter%205_html_m17b64539.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6942/Chapter%205_html_m319891f6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6942/Chapter%205_html_193f9bb9.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6942/Chapter%205_html_771ab082.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6942/Chapter%205_html_3ad2122b.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6942/Chapter%205_html_m3adb796c.gif)
From (1), (2), and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6942/Chapter%205_html_m3d78aab0.gif)
Question 11:
Differentiate the function with respect to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6949/Chapter%205_html_3e605c11.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6949/Chapter%205_html_6064c284.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6949/Chapter%205_html_m59e0dc54.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6949/Chapter%205_html_33ac2b2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6949/Chapter%205_html_m6800765d.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6949/Chapter%205_html_760c0f18.gif)
From (1), (2), and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6949/Chapter%205_html_md0b5321.gif)
Question 12:
Find
of function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6955/Chapter%205_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6955/Chapter%205_html_3b66b7c4.gif)
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6955/Chapter%205_html_3b66b7c4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6955/Chapter%205_html_3b66b7c4.gif)
Let xy = u and yx = v
Then, the function becomes u + v = 1
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6955/Chapter%205_html_m6e2ab7d2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6955/Chapter%205_html_m58067ee2.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6955/Chapter%205_html_m2c850875.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6955/Chapter%205_html_m32442ecb.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6955/Chapter%205_html_m34966692.gif)
From (1), (2), and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6955/Chapter%205_html_m51d87f3b.gif)
Question 13:
Find
of function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6959/Chapter%205_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6959/Chapter%205_html_m243ff079.gif)
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6959/Chapter%205_html_m243ff079.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6959/Chapter%205_html_m243ff079.gif)
Taking logarithm on both the sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6959/Chapter%205_html_m586a78c.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6959/Chapter%205_html_1c72ea1e.gif)
Question 14:
Find
of function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6964/Chapter%205_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6964/Chapter%205_html_m50f80e68.gif)
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6964/Chapter%205_html_m50f80e68.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6964/Chapter%205_html_m50f80e68.gif)
Taking logarithm on both the sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6964/Chapter%205_html_61526371.gif)
Differentiating both sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6964/Chapter%205_html_7bed3156.gif)
Question 15:
Find
of function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6972/Chapter%205_html_m72221781.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6972/Chapter%205_html_m44b33473.gif)
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6972/Chapter%205_html_m44b33473.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6972/Chapter%205_html_m44b33473.gif)
Taking logarithm on both the sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6972/Chapter%205_html_m6648756a.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6972/Chapter%205_html_14891e0e.gif)
Question 16:
Find the derivative of the function given by
and hence find
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6975/Chapter%205_html_m7709a18f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6975/Chapter%205_html_13a18c98.gif)
Answer:
The given relationship is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6975/Chapter%205_html_m7709a18f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6975/Chapter%205_html_m7709a18f.gif)
Taking logarithm on both the sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6975/Chapter%205_html_5394fe11.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6975/Chapter%205_html_3196b276.gif)
Page No 179:
Question 18:
If u, v and w are functions of x, then show that
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6977/Chapter%205_html_64c95fbc.gif)
in two ways-first by repeated application of product rule, second by logarithmic differentiation.
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6977/Chapter%205_html_m3be4ac90.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6977/Chapter%205_html_m3be4ac90.gif)
By applying product rule, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6977/Chapter%205_html_m2058d63e.gif)
By taking logarithm on both sides of the equation
, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6977/Chapter%205_html_22bb0e2f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6977/Chapter%205_html_m76a64de7.gif)
Differentiating both sides with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6977/Chapter%205_html_e86d78d.gif)