NCERT Solutions for Class 11 Maths Chapter 13 – Limits and Derivatives Ex 13.2
Page No 312:
Question 1:
Find the derivative of x2 – 2 at x = 10.
Answer:
Let f(x) = x2 – 2. Accordingly,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4895/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m2f2d2f4f.gif)
Thus, the derivative of x2 – 2 at x = 10 is 20.
Question 2:
Find the derivative of 99x at x = 100.
Answer:
Let f(x) = 99x. Accordingly,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4896/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_2ffb6971.gif)
Thus, the derivative of 99x at x = 100 is 99.
Question 3:
Find the derivative of x at x = 1.
Answer:
Let f(x) = x. Accordingly,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4897/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7158fa45.gif)
Thus, the derivative of x at x = 1 is 1.
Question 4:
Find the derivative of the following functions from first principle.
(i) x3 – 27 (ii) (x – 1) (x – 2)
(ii)
(iv) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4898/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_62e09733.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4898/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_170df687.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4898/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_62e09733.gif)
Answer:
(i) Let f(x) = x3 – 27. Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4898/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_2d6d2d35.gif)
(ii) Let f(x) = (x – 1) (x – 2). Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4898/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_22455bea.gif)
(iii) Let
. Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4898/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m5cbbab3b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4898/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_aebf2ac.gif)
(iv) Let
. Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4898/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_612efa8f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4898/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_67857ef2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4898/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m9f86c84.gif)
Question 5:
For the function
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4899/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_44bd0b1b.gif)
Prove that ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4899/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1b489393.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4899/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1b489393.gif)
Answer:
The given function is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4899/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_69770863.gif)
Thus,![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4899/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_7a105d4b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4899/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_7a105d4b.gif)
Page No 313:
Question 6:
Find the derivative of
for some fixed real number a.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4900/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_4c51c46.gif)
Answer:
Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4900/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_41cedd6f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4900/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_41cedd6f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4900/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_bcea835.gif)
Question 7:
For some constants a and b, find the derivative of
(i) (x – a) (x – b) (ii) (ax2 + b)2 (iii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4901/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1ecb723f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4901/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m1ecb723f.gif)
Answer:
(i) Let f (x) = (x – a) (x – b)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4901/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_1b5b786a.gif)
(ii) Let ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4901/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_1737dc47.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4901/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_1737dc47.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4901/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m3e620e37.gif)
(iii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4901/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_12f78103.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4901/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_12f78103.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4901/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_651e7f03.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4901/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_1303c565.gif)
Question 8:
Find the derivative of
for some constant a.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4902/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m47d881c.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4902/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_7eff7f86.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4902/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m7fb02060.gif)
Question 9:
Find the derivative of
(i)
(ii) (5x3 + 3x – 1) (x – 1)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m5acefce2.gif)
(iii) x–3 (5 + 3x) (iv) x5 (3 – 6x–9)
(v) x–4 (3 – 4x–5) (vi) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_38b8768c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_38b8768c.gif)
Answer:
(i) Let![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m74a48404.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m74a48404.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m6853c5f7.gif)
(ii) Let f (x) = (5x3 + 3x – 1) (x – 1)
By Leibnitz product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_3ee3c61c.gif)
(iii) Let f (x) = x– 3 (5 + 3x)
By Leibnitz product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m54d06c0f.gif)
(iv) Let f (x) = x5 (3 – 6x–9)
By Leibnitz product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_50f7d65.gif)
(v) Let f (x) = x–4 (3 – 4x–5)
By Leibnitz product rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m3c6fa350.gif)
(vi) Let f (x) = ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_38b8768c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_38b8768c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m3756297c.gif)
By quotient rule,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4903/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_2b7bf93f.gif)
Question 10:
Find the derivative of cos x from first principle.
Answer:
Let f (x) = cos x. Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4904/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m37e6bef4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4904/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m4c9af0a5.gif)
Question 11:
Find the derivative of the following functions:
(i) sin x cos x (ii) sec x (iii) 5 sec x + 4 cos x
(iv) cosec x (v) 3cot x + 5cosec x
(vi) 5sin x – 6cos x + 7 (vii) 2tan x – 7sec x
Answer:
(i) Let f (x) = sin x cos x. Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4905/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_304403a7.gif)
(ii) Let f (x) = sec x. Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4905/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m5f9ef6fd.gif)
(iii) Let f (x) = 5 sec x + 4 cos x. Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4905/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_7ed90d33.gif)
(iv) Let f (x) = cosec x. Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4905/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_62ad7951.gif)
(v) Let f (x) = 3cot x + 5cosec x. Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4905/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m2af50ac5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4905/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_2f846c3c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4905/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m511e6840.gif)
From (1), (2), and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4905/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m5ebf6e66.gif)
(vi) Let f (x) = 5sin x – 6cos x + 7. Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4905/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m5a07b0f4.gif)
(vii) Let f (x) = 2 tan x – 7 sec x. Accordingly, from the first principle,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/173/4905/NCERT_16-10-08_Khushboo_11_Math_Ex-13.1_32_SU_SNK_html_m59906960.gif)