NCERT Solutions for Class 11 Maths Chapter 13 – Limits and Derivatives Miscellaneous Exercise
Page No 317:
Question 1:
Find the derivative of the following functions from first principle:
(i) –x (ii) (–x)–1 (iii) sin (x + 1)
(iv) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_20d7320.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_20d7320.gif)
Answer:
(i) Let f(x) = –x. Accordingly,![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_6e76cb37.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_6e76cb37.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_5826e1ad.gif)
(ii) Let
. Accordingly,![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_d2b3b7b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_58fc58da.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_d2b3b7b.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_e210ce8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_345f75de.gif)
(iii) Let f(x) = sin (x + 1). Accordingly,![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_m2c5141b1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_m2c5141b1.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_m686d2989.gif)
(iv) Let
. Accordingly,![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_36a47e48.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_m2d506601.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_36a47e48.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_63665a00.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5634/Chapter%2013%20Math_html_291566ce.gif)
Question 2:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + a)
Answer:
Let f(x) = x + a. Accordingly,![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5117/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1e86568f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5117/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1e86568f.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5117/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_2464327f.gif)
Question 3:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5118/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m73f4bba2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5118/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m73f4bba2.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5118/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_46e5fc4f.gif)
By Leibnitz product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5118/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m6d52e72e.gif)
Question 4:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b) (cx + d)2
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5119/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_631ab596.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5119/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_631ab596.gif)
By Leibnitz product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5119/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_ffa1e6.gif)
Question 5:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5120/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m67ef0957.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5120/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m67ef0957.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5120/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7aefe851.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5120/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7aefe851.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5120/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m2b654dad.gif)
Question 6:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5121/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m68f13af0.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5121/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_1563ba01.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5121/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_5d6c81b.gif)
Question 7:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5122/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_mccfaf22.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5122/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_mccfaf22.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5122/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m2d2cbf30.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5122/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m2d2cbf30.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5122/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m62439bda.gif)
Question 8:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5123/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m195536bf.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5123/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m195536bf.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5123/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1871e4a0.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5123/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m49a81c44.gif)
Question 9:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5124/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7c209ad0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5124/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7c209ad0.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5124/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7c898831.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5124/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_5a1013fb.gif)
Question 10:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5125/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m39576295.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5125/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m39576295.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5125/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m5c866e32.gif)
Question 11:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5126/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_54da3b0a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5126/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_54da3b0a.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5126/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_328cc96d.gif)
Question 12:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b)n
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5127/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_3a014f96.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5127/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7581261.gif)
Question 13:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b)n (cx + d)m
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5128/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_3ea80f0e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5128/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_3ea80f0e.gif)
By Leibnitz product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5128/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_6c548342.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5128/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m6c641c13.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5128/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_5be27cbc.gif)
Therefore, from (1), (2), and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5128/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m24291f42.gif)
Question 14:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sin (x + a)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5129/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_325f110d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5129/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_325f110d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5129/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1596c910.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5129/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_52e709df.gif)
Question 15:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): cosec x cot x
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5130/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7893d445.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5130/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7893d445.gif)
By Leibnitz product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5130/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_24972f04.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5130/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_1cbdf2a5.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5130/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m5d515034.gif)
Now, let f2(x) = cosec x. Accordingly,![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5130/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7a3164e8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5130/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7a3164e8.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5130/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_5d9fefac.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5130/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m5dcb0c86.gif)
From (1), (2), and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5130/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1b8b4bf5.gif)
Question 16:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5131/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m2cf197c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5131/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m2cf197c.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5131/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_5c0f547d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5131/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_5c0f547d.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5131/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m2a36af34.gif)
Page No 318:
Question 17:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5132/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1e0c0678.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5132/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1e0c0678.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5132/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_23158b33.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5132/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_23158b33.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5132/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m68b172a3.gif)
Question 18:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5133/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_68edaf34.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5133/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_68edaf34.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5133/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_6a8b4a1b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5133/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_6a8b4a1b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5133/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7b829b60.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5133/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m58da9801.gif)
Question 19:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sinn x
Answer:
Let y = sinn x.
Accordingly, for n = 1, y = sin x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5134/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m67feca5d.gif)
For n = 2, y = sin2 x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5134/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_77223e3.gif)
For n = 3, y = sin3 x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5134/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1bcbde70.gif)
We assert that ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5134/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_6f914708.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5134/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_6f914708.gif)
Let our assertion be true for n = k.
i.e., ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5134/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7fb25575.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5134/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7fb25575.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5134/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_5223811f.gif)
Thus, our assertion is true for n = k + 1.
Hence, by mathematical induction,![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5134/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_6f914708.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5134/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_6f914708.gif)
Question 20:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5135/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_mfc656ae.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5135/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_mfc656ae.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5135/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_31b92683.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5135/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_20d501d7.gif)
Question 21:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5136/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1df3334c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5136/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1df3334c.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5136/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_2e407a5d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5136/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_2e407a5d.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5136/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_7cd5575e.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5136/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_1ab9e173.gif)
From (i) and (ii), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5136/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m74f81f1a.gif)
Question 22:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x4 (5 sin x – 3 cos x)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5137/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m78cc85ed.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5137/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m78cc85ed.gif)
By product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5137/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_2348a029.gif)
Question 23:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x2 + 1) cos x
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5138/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m5d3f8ee6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5138/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m5d3f8ee6.gif)
By product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5138/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_73cc370c.gif)
Question 24:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax2 + sin x) (p + q cos x)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5139/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_630f99ac.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5139/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_630f99ac.gif)
By product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5139/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m33ca1fcf.gif)
Question 25:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5140/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m38159b81.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5140/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m38159b81.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5140/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_2d814833.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5140/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_2d814833.gif)
By product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5140/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_5c68adf7.gif)
Let
. Accordingly,![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5140/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m41df1355.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5140/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m767a4c32.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5140/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m41df1355.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5140/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m4aa2503d.gif)
Therefore, from (i) and (ii), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5140/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m4023f64e.gif)
Question 26:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5141/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m6da65dc3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5141/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m6da65dc3.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5141/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_6a68f69f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5141/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_6a68f69f.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5141/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_4bcb8a1d.gif)
Question 27:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5142/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_127cf76e.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5142/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_620d65c8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5142/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_620d65c8.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5142/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_787ee1c0.gif)
Question 28:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5143/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_4512493f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5143/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_4512493f.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5143/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_175c4230.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5143/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_175c4230.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5143/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m13aff7c4.gif)
By first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5143/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_527fd9cf.gif)
From (i) and (ii), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5143/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_38a7aa52.gif)
Question 29:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + sec x) (x – tan x)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5144/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m50843966.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5144/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m50843966.gif)
By product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5144/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_794ea5c1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5144/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_45f3614e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5144/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7c3e755c.gif)
From (i), (ii), and (iii), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5144/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_484c8a81.gif)
Question 30:
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5145/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_4f2cc197.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5145/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_4f2cc197.gif)
Answer:
Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5145/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_5588c26a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5145/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_5588c26a.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5145/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m4c8c47eb.gif)
It can be easily shown that ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5145/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_1ca7d57b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5145/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_1ca7d57b.gif)
Therefore,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/5145/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_mf33201d.gif)