NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability Ex 5.7
Page No 183:
Question 1:
Find the second order derivatives of the function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6990/Chapter%205_html_m7a59faf5.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6990/Chapter%205_html_41f03fdb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6990/Chapter%205_html_41f03fdb.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6990/Chapter%205_html_140e559d.gif)
Question 2:
Find the second order derivatives of the function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6991/Chapter%205_html_m2468ff53.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6991/Chapter%205_html_m7bd30f86.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6991/Chapter%205_html_m7bd30f86.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6991/Chapter%205_html_m57b41e71.gif)
Question 3:
Find the second order derivatives of the function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6992/Chapter%205_html_m23f4c69b.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6992/Chapter%205_html_2aec51b6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6992/Chapter%205_html_2aec51b6.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6992/Chapter%205_html_71a5f565.gif)
Question 4:
Find the second order derivatives of the function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6993/Chapter%205_html_727b0d3e.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6993/Chapter%205_html_m5d5bb1c3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6993/Chapter%205_html_m5d5bb1c3.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6993/Chapter%205_html_4c379a84.gif)
Question 5:
Find the second order derivatives of the function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6994/Chapter%205_html_33c4eae6.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6994/Chapter%205_html_m34818b33.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6994/Chapter%205_html_m34818b33.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6994/Chapter%205_html_192f1554.gif)
Question 6:
Find the second order derivatives of the function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6995/Chapter%205_html_480384f8.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6995/Chapter%205_html_m338bf3ed.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6995/Chapter%205_html_m338bf3ed.gif)
Then, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6995/Chapter%205_html_mf68e5a3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6995/Chapter%205_html_mf68e5a3.gif)
Question 7:
Find the second order derivatives of the function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6996/Chapter%205_html_47bf6e60.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6996/Chapter%205_html_6f6eedb7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6996/Chapter%205_html_6f6eedb7.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6996/Chapter%205_html_m72401aa9.gif)
Question 8:
Find the second order derivatives of the function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6997/Chapter%205_html_6b1d3f6d.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6997/Chapter%205_html_m5f1b07e9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6997/Chapter%205_html_m5f1b07e9.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6997/Chapter%205_html_m1e4696a0.gif)
Question 9:
Find the second order derivatives of the function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6998/Chapter%205_html_f8f0eb6.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6998/Chapter%205_html_6451572a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6998/Chapter%205_html_6451572a.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6998/Chapter%205_html_m49c47554.gif)
Question 10:
Find the second order derivatives of the function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6999/Chapter%205_html_m4ebb87cd.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6999/Chapter%205_html_5f4c3a91.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6999/Chapter%205_html_5f4c3a91.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6999/Chapter%205_html_203b096f.gif)
Question 11:
If
, prove that![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7000/Chapter%205_html_m2b67ca89.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7000/Chapter%205_html_m287fb11e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7000/Chapter%205_html_m2b67ca89.gif)
Answer:
It is given that,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7000/Chapter%205_html_m287fb11e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7000/Chapter%205_html_m287fb11e.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7000/Chapter%205_html_4a272fe7.gif)
Hence, proved.
Page No 184:
Question 12:
If
find
in terms of y alone.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7001/Chapter%205_html_m7285ceb0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7001/Chapter%205_html_48519442.gif)
Answer:
It is given that,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7001/Chapter%205_html_27f47e6b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7001/Chapter%205_html_27f47e6b.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7001/Chapter%205_html_500433ae.gif)
Question 13:
If
, show that![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7002/Chapter%205_html_m6fb302b5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7002/Chapter%205_html_6a6aad23.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7002/Chapter%205_html_m6fb302b5.gif)
Answer:
It is given that,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7002/Chapter%205_html_6a6aad23.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7002/Chapter%205_html_6a6aad23.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7002/Chapter%205_html_m1d04d99e.gif)
Hence, proved.
Question 14:
If
show that![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7003/Chapter%205_html_mdce2140.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7003/Chapter%205_html_3538e479.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7003/Chapter%205_html_mdce2140.gif)
Answer:
It is given that,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7003/Chapter%205_html_m3ce836af.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7003/Chapter%205_html_m3ce836af.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7003/Chapter%205_html_m1a6b17e8.gif)
Hence, proved.
Question 15:
If
, show that![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7005/Chapter%205_html_m61d13575.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7005/Chapter%205_html_m173671fd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7005/Chapter%205_html_m61d13575.gif)
Answer:
It is given that,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7005/Chapter%205_html_m173671fd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7005/Chapter%205_html_m173671fd.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7005/Chapter%205_html_m12fbda7f.gif)
Hence, proved.
Question 16:
If
, show that![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7006/Chapter%205_html_m365ab33e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7006/Chapter%205_html_m1f2a0c9e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7006/Chapter%205_html_m365ab33e.gif)
Answer:
The given relationship is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7006/Chapter%205_html_m1f2a0c9e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7006/Chapter%205_html_m1f2a0c9e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7006/Chapter%205_html_4513d279.gif)
Taking logarithm on both the sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7006/Chapter%205_html_m7a51c783.gif)
Differentiating this relationship with respect to x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7006/Chapter%205_html_4686e8e2.gif)
Hence, proved.
Question 17:
If
, show that![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7007/Chapter%205_html_m616df3d0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7007/Chapter%205_html_362aaa15.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7007/Chapter%205_html_m616df3d0.gif)
Answer:
The given relationship is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7007/Chapter%205_html_362aaa15.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7007/Chapter%205_html_362aaa15.gif)
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/7007/Chapter%205_html_m10ec30e3.gif)
Hence, proved.