Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Miscellaneous Exercise
Limits and Derivatives Miscellaneous Exercise Solutions
1. Find the derivative of the following functions from first principle:
(i) –x
(ii) (–x)–1
(iii) sin (x + 1)
(iv) cos(x - π/8)
Solution
(i) Let f(x) = -x . Accordingly, f(x + h) = -(x + h)
By first principle,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjstv1GibreIJDfFdcCecjf-E4UIXei_-QCthjRrzEbyNYzdHIVmfcq7D96IiNtTu4PD61NxH_BgbixP5OFBn9oGDWivu4t9GzXdjLKLGLkuGWiMnMv_nVpiB07iYUo7MInPg6uMeFZfJBL5rh-yLlA4o7TgwFndN2jl7TADoJmUQ_Svxrcp14H-16k/w198-h225-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%201.JPG)
(ii) Let f(x) = (-x)–1 = 1/-x = -1/x . Accordingly, f(x + h) = -1/(x + h)
By first principle,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgf9rC_HKMb1EhdVG-zVHbgh8gt-iaRc4Ox33Jcep2LecwsYdyLwvTi0VMxyvjlTbZqXQeMMNz_v62vD2dWKZQ8tQs4nfZ23w7Q6yshAu_S70zKa8u5ChzkzFgu-tR1FbhxfTkwrwCm1sBnguC80woSlmfiZwNv3iXlmBYyZ05O3itJANXgRI1_hmth/w189-h443-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%202.JPG)
(iii) Let f(x) = sin (x + 1). Accordingly, f(x + h) = sin(x + h+ 1)
By first principle,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7aHYf-gcVMMD44m7G7vpH4X91EgowQYboIFjTn38JBcGNJHxZMZ9wjSZ7ahmOazxvWzw5dcYhOOXOlcRixLDdXtVA1rcH28wSSPtkxlCkORe5BklnEJPaDVw20OfydzfELwyQ_Nj7PLmCFQJ_0hO2UWAsL38tg3ovT_zI9aH9bEXwn33fso7wM6Hr/w392-h418-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%203.JPG)
= cos(x + 1)
(iv) Let f(x) = cos (x - π/8). Accordingly, f(x + h) = cos (x + h - π/8)
By first principle,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVSMZ-lmSB2HkNaplAPPs4Ic8FPk0xX3QxgdPVWw0V4wf1_TJJwQxU4zOuI36Xk-CFPEy-PaU6YclkBcqupNVSjAK9zGTgE_y0o3AlUNvO-mfrjiLKcszWKtPjqEb3Ne1kRzRndAZpHAB5bGdEbMUwim5urMwVQgsMCvwMQr5aTM1QKfSZS-a9-Jtu/w406-h571-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%204.JPG)
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)
Solution
Let f(x) = x + a. Accordingly, f(x + h) = x + h + a
By first principle,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjanprfvPZNE7eFpGAK37g0iMSnZLHcu5HE82zuNO71MJOPs-QnPJQkvi8DlOTr04Qp9J9HNt5F5Q1gFaHWVZp4hsMqroerXNE5GUX_RtwHP-fDa9wz69IJxNsuCdsDjRDiQRo6GgHFRmCss_j_Ry2vLC3HM_gxcUGmmGTloPoJV3O-zQ5UaZ_6_4n3/w188-h187-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%205.JPG)
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): (px + q)(r/s + s)
Solution
Let f(x) = (px + q)(r/s + s)
By Leibnitz product rule,
4. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers): (ax + b) (cx + d)2 .
Solution
Let f(x) = (ax + b)(cx + d)
2 By Leibnitz product rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ0dcO0qxkjsdNgCx6wvfMIW4Cz7Bvy-AZy8hraIsQPuYascGlcVfdip2naayf6kukOtukjEK0yhCa59kXZVzWAHLbL_BvkBVwkY9dYp1FPtq3Qbxnn1KdJR-lRhiCNnwPoJfba8oNy8w_ZgWvd34jefNJNdnL5RBWOkr5ilLrTLsMmb3c-dXdJPAq/w480-h147-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%207.JPG)
= (ax + b)(2c
2 x + 2cd) + (cx + d
2)a
= 2c(ax + b)(cx + d) + a(cx + d)
2
5. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers): (ax + b)/(cx + d).
Solution
Let f(x) = (ax+ b)/(cx + d)
By quotient rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9xIUuPt6DuEy0DdZ71iMTGXoMC9mWvaiXZHbfZr5DZYqW14UWGJOwGx5K-zjuCdpfkCa1d-_cJinoHuzo6scD4SDcymO07APxR8Rn3NDNwIsZFuzQ-3Tvs9K5LD7Q37kU9g4Ok6O2WgOGkzaBtSz8f7Ofop8MWVEouarw6Wm0NFXOWh7LRyDwPFmA/w340-h228-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%208.JPG)
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): (1 + 1/x)/(1 - 1/x).
Solution
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): 1/(ax2 + bx + c) .
Solution
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) : (ax + b)/(px2+ qx + r) .
Solution
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): (px2 + qx + r)/(ax + b)
Solution
10. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers) : a/x4 = (b/x2) + cos x
Solution
11. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers): 4√x – 2 .
Solution
Let f(x) = 4√x – 2
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGtCzY3XOXb2dUy5CyVdoKOkdI8qpxsCV8WtWvMMvrSQEKunfOtaauFtVB0ogZhMTdiyrAs0dcFSLhBu9JcUwUSkseYOs34xPMT3KStmsgpMqK1LY0ltbBwV5HOsPEby5M-NCjWkB6C32Ocyr5sAXCt0G1JTXZQAYxrSOY-Y559WhzVcIz8O59Y7HX/w296-h155-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2014.JPG)
12. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers): (ax + b)n .
Solution
Let f(x) = (ax + b)
n . Accordingly, f(x + h) = {a(x + h) + b}
n = (ax + ah + b)
n By first principle,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHGbcgnj8B0nEkeulJhMRRYefgDSNGjJvipynKJa2wKIvvqP7YpIR5VqFTUpJc49cRyD7vYiWCJ-GFHfavCaEB25ZI-gvSI-KRF-UcSW4Y6Z8ytudKryltjAQ52c_4PeENiGSmkvCR3KNwz9tY33fTUOGQg3EkeFR7XuS_H2yQmK4r5-ceDGI2exXT/w558-h559-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2015.JPG)
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
Solution
Let f(x) = (ax + b)
n (cx + d)
m By Leibnitz product rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJumiCQM8HrPvnrSzsml0ATYu_3n4vvPkSJLr0ojw_vyEocuy1AdERD4SSYIXeUVnbSXCRxIYIcKf_yKGyFHtvQRcESmVpc8dsmSkKjxMdTI2E2lPscv3IXmQ2TzAFXHPJ77R7o_diIuasOQOoiKD5VfgiYUrRISoiDMfRmwIL5Lm1DdkSXu8Z1nLN/w577-h668-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2016.JPG)
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)
Solution
Let f(x) = sin (x + a)
f(x + h) = sin (x + h + a)
By first principle,
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
Solution
Let f(x) = cosec x cot x
By Leibnitz product rule,
f'(x) = cosec x (cot x)' + cot x(cosec x)' ...(1)
Let f1 (x) = cos x. Accordingly, f1 (x + h) = cot (x + h)
By first principle,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlguPp4GOuMPT40ZtJ8t_yETu5sRDN9FHPYhzCng7cOMUy9rkGdhnTzGNyJsBSc5o7GdcdBFRTTJc26FhMbUH0Q_pdy9Elg0uNRnf0GxxdOnpOkUPo57E7qtq_0HA_10nvC4ezwSGsfrb4yIRztKOt_-Ue70i1aG5xYEv9bUdatgRX5KVCa6dIVarl/w326-h553-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2018.JPG)
= - cosec
2 x
∴ (cot x)' = -cosec
2 x
...(2) Now, let f
2 (x) = cosec x . Accordingly, f
2 (x + h) = cosec (x + h)
By first principle,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWFzJsJTHjJtzLWvMoVMzD73jzYHzzyZ9-fKrruf5fD8WK6fT_JbRQ0EUpAjY4mHZ_GglQ8BvHt7NNnYz4IrPEm7BdnF6rwLvSVmh5aOXIbTszjPbgvpbp0xjMqFVTy5SGDqj9atgIsyZjZiI763z3EBOdPdYx8PisqKih_d8Eh5xoz17ZRF2ju7NR/w318-h690-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2019.JPG)
= - cosec x . cot x
∴ (cosec x)' = -cosec x. cot x
...(3) From (1), (2), and (3), we obtain
f '(x) = cosec x(- cosec
2 x) + cot x( -cosec x cot x)
= -cosec
3 x - cot
2 x cosec x
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):
Solution
Let f(x) = cos x/(1 + sin x)
By quotient rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOAqOT47mVI8N0GyPJXcT4QhKY23OMirIZ3GKO6H1EsaOxp4Bu6cm-CJE2---NHbxTNb6TjLy-1ve8BwmNpGRtn_PAoDrTHI2RWxOofFDkKP5FAXJ4vI4y9fvtLWdph_lPms6jPA3S7XSgSkQ-11ZyI6-Nq9wacH_OrSK7a2q9231ayclkAiXmj7T-/w331-h382-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2020.JPG)
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) : (sin x + cos x)/(sin x - cos x)
Solution
Let f(x) = (sin x + cos x)/(sin x - cos x)
By quotient rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgD8GV6VJ2BxY_0pD7d5ivrmTK0908Rd5FXDBT0zj4J83t8i_vU4O2qunpYIZT9uE6DXwKerMG3UUYiEXMnfT44ZqRGayX_mS0vkvJ8CIom3tkQpZDTQERDOvCV1JFsFEDuwPrMiaKXXyqCleO8ROlLmpfSMOwI8ISPD_8n-blCZJl9GKwJV4MPVOiv/w472-h341-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2021.JPG)
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) : (sec x - 1)/(sec x + 1)
Solution
Let f(x) = (sec x - 1)/(sec x + 1)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUw-0hXZxo0Ai4Kr9LytA0Fvq0wDgzRYvv3dIKqq-4-Al0kkuNe8Uvskik-Dz9OmKS6cIwB0_i0K3G9UiMCfGacg2HrZOFrjiEbITxC30xgXSaNihFd648bx8JlSGZY68UkGObAkuvvV7VIPb36VU0-ST4idB-AMf0PPlYOOhcDaypfWBbzmaJPMQO/w356-h553-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2022.JPG)
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
Solution
Let y = sinn x.
Accordingly, for n = 1, y = sin x.
∴ dy/dx = cos x, i.e., (d/dx) sin x = cos x
For n = 2, y = sin2 x.
∴ dy/dx = (d/dx) (sin x sin x)
= (sin x)' sin x + sin x(sin x)'
[By Leibnitz product rule] = cos x sin x + sin x cos x
= 2 sin x cos x
...(1) For n = 3, y = sin
3 x .
∴ dy/dx = (d/dx)(sin x sin
2 x)
= (sin x)' sin
2 x + sin x(sin
2 x)'
[By Leibnitz product rule] = cos x sin
2 x + 2 sin
2 x cos x
= 3 sin
2 x cos x
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuHUi-6tLR7LAWDe2L6qi09Z6J0td3qPPoYq4kzEpzEvH2ujqYIWXDmCD2QIF0EAYLu9tbaGlo2JW91JItwrPsF3vIKLiglXetKsNJRrnllROx6OrDT4GRMK7DBR79IxjVEJRjMPkBGq3jEQrQAOjrwloWwUvDnVElUpscfSD6wlSUGdndyTjolxaB/w291-h176-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2023.JPG)
= (sin x)' sin
k x + sin x(sin
k x)'
[By Leibnitz product rule]= cos x sin
k x + sin x(k sin
(k-1) x cos x)
[Using (2)] = cos x sin
k x + k sin
k x cos x
= (k + 1)sin
k x cos x
Thus, our assertion is true for n = k + 1
Hence, by mathematical induction , d/dx (sinn x) = n sin(n-1) x cos x
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): (a + b sinx)/(c + d cos x)
Solution
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): [sin (x + a)]/cos x .
Solution
Let f(x) = [sin (x + a)]/cos x
By quotient rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3OAzA9kDbNqcOK2omN-9tydtDuw_UW1wyJpSQX4-iNjI-eUmv7-dsMN924wkvEu4hWFMkJncjYgLgfUby4YclO2Y1r4NVhBqiwVAJXLjSjvRRf91ojT-6Yy6GpPrYN--_XC9Ld7JAI2OPgNMXMNJoh5dpQK-5XlClCCjQsvObm9Luwvq0VUS6mO-R/w448-h133-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2025.JPG)
Let g(x) = sin (x + a). Accordingly, g(x + h) = sin (x + h + a)
By first principle,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguMWHMua1jTwZPOcAVZxplQQdKPME20OfgPUXC2U1MaJDYx8nJ-_HlKSQvy-POhjCphXaB93tJJo9a7TpwZ0OxjXt4YZfSLzAyCLZTmz79rT-diHQUtykw68t0HbCqOyvGUiVhsUGlKCDIjXGE-94cW59XdORQxMIEuoest2a13356v9XOivV4Ti_S/w441-h421-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2026.JPG)
= cos (x + a)
...(ii) From (i) and (ii), we obtain
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWmIt9mggEzPPnIDvUFsyfCjTVMjgwf70Mbm5zll6mzJgkbqxOO7nA5yOGRhqNlfh21qGTPezTWHEfFIE2cN4EHc60mHL1Tec-5Qj7Z6XG-0vmxabMzOf1SYZDCE3zu1CekCxwWEAXNHwWfYr0LbhGdpKhWV32uekfZ_hvx7Gem8Vg2FxIgshRuY_e/w270-h131-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2027.JPG)
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)
Solution
Let f(x) = x4 (5 sin x – 3 cos x)
By product rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioWgVUHrJB29u15-hPpnKA5zc0HzSd6EwznlNW7bhjd-HjBm6tWbCIGYjBAKsmiVenXOHK7y08x14Hu5W1xwqtxXzuD8VRy3xEi56vY4qcwAe45SxFQ3tM5uByndCpumVmv3NUVAU0tBVBlFKSKVpWonHZQ4v1zZY8D0l8FytmMQwiKtfI5GUFRDGJ/w397-h153-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2028.JPG)
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
Solution
Let f(x) = (x
2 + 1) cos x
By product rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinH1jZdm0Tt1iuCEwmIwgh8tK8MxTnuQ-FjLmX7Y5hgEWE3NR4T-X5ztQyZbLjkJPhafCzGsOh4NZTcsWSLO8H8Pe5vADgHywQj_nleliGZW0VtA-SZs1A-PaOngyWIM00nagr35RjGBOM5Z8CV6PdImXzdIy6_ahr8nN2jJ0RjhCOfqM8iHSRfqCy/w329-h47-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2029.JPG)
= (x
2 + 1)(- sin x) + cos x(2x)
= -x
2 sin x - sin x + 2x cos x
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)
Solution
Let f(x) = (ax
2 + sin x)(p + q cos x)
By product rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjX6j1V5Er-oYzYYX5jOVpPyUVJmQNdgDpe3kiRTIinpm8wUBJobw2D_LUevhLg3SealnSrN9ffhEJL8dnkANMAV-7Q4ra_36HcZGXUu1vuLyp8H7yJZiiz4hXkcqBNh7koeOhhd07KlEmFSeQZMav_ANllkekbjSODHz5d2nc4fWrbx-O6z-sEfSjI/w463-h46-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2030.JPG)
= (ax
2 + sin x)(-q sin x) + (p + q cos x)(2ax + cos x)
= -q sin x (ax
2 + sin x) + (p + q cos x) (2ax + cos x)
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): (x + cos x)(x - tan x)
Solution
Let f(x) = (x + cos x)(x - tan x)
By product rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZKOw8T2KFlrC5ts7zpm65ACNbCSuQKy79tYFj90Gh9tK-zXP9AFFl-3rzxytb21CmpxF6_f95JMoYXGTtTJto4Z3_hf6l9-_bOh3xw9QuXH8WQ30nh6KqQY4LXCvsUjszdRl9YWaXdPyzN1FcUJ_qhXECzVADW-9otJF_HAZhAf2sEopSggAWosb4/w434-h141-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2031.JPG)
Let g(x) = tan x. Accordingly, g(x + h) = tan (x + h)
By first principle,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuPSKRGKLAuaTBL5iuJwt6Im6qoeWdaAd5sQXBMjifbkCD-hyL7tuv4i1gIfk7qXZG88-pAtv1PnTF3Q0-eG-xwORmmaEK06nG4dq6dcXQbqzSpJSLz_gnwtUv97Sb53P5JS_k2VQAKc2Zj0vh8ZnSo6H74LjbuyDyaM6Br1meCUVLYgDehHVShlTu/w275-h461-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2032.JPG)
= sec
2 x
...(ii) Therefore, from (i) and (ii), we obtain
f '(x) = (x + cos x)(1 - sec
2 x) + (x - tan x)(1 - sin x)
= (x + cos x)(-tan
2 x) + (x - tan x)(1 - sin x)
= -tan
2 x (x + cos x) + (x - tan x) (1 - sin x)
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):
Solution
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):
Solution
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): x/(1 + tan x)
Solution
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEju_nlNVlZsthU-9LF_-bfr05hJKN2b8uQ5N1G1EUDoJVHt_T9nXVizmc3_UWPQjRAoN9l3QpbFludB_x_1jI8jBNBbLudur4rZ8zs8tAArYH_7OWRW1o9O84HI06HEw4n5FGj1hiHqBcV4kjFN21qThl05ul0X4ELkiTGr8y9BDhdutMAPXIdrP2aZ/w373-h191-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2035.JPG)
Let g(x) = 1 + tan x. Accordingly, g(x + h) = 1 + tan (x + h).
By first principle,
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)
Solution
Let f(x) = (x + sec x)(x -tan x)
By product rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRVKgc7ozDoQMPDykx0GOGNTIOn7SMqr7qdmd_xCowtInXHVA4y7GCcQShRNX0dHRVQowgPMjN_vKeApFVURE_BZgtMQ02vexYGV7amAB41WmnOEt3Vwjx0uwoDUPBKXbteRWBtLAKBgByNJoEX30LqDSXl1XZUnH_HZ9EERWGt3g0zOHJCwDeaAxo/w476-h132-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2037.JPG)
Let f
1 (x) = tan x, f
2 (x) = sec x
Accordingly, f
1 (x + h) = tan (x + h) and f
2(x + h) = sec (x + h)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAyQj3LhGfJXSA3mEwDe6BUcD5IUICOhL1Xd9IITCYkuXoETv48qC7uhy3lPZ3ZLxwik5LtXAlyxDWjibdT7Eg3u9CzqWAmgDYd2Fgg-SXs7dM2Uwvn0TgkuXVgmRMYBUHttNar5drBWPDf_PMjtU89L-eAWgKYavHUcsXFFEJK5SMFE_awZhrPu0l/w344-h1322-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2038.JPG)
⇒ (d/dx) sec x = sec x tan x
...(iii) From (i), (ii) and (iii), we obtain
f'(x) = (x + sec x) (1 - sec
2 x) + (x - tan x)( 1 + sec x tan x)
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): x/sinn x
Solution
Let f(x) = x/sin
n x
By quotient rule,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNUrBWiNqLIUA7Q0Be9rFZCM0Gu0S8jveVIo8i27LuLBpagIPsaqhnuJuhnv3O-9UJWPp7jb7e4FvyqG0L75XvlnBPQmflQKsEa4_IKBpoSQ897qq0XNXm3oS9sxCHvvekl0itPPtt_7LFcN7osZ7jicFLhPB8tEHnDnFiwuvM9A05rqZzdYsQnPRz/w392-h369-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Miscellaneous%20Exercise%20img%2039.JPG)