--Advertisement--

Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Miscellaneous Exercise

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, 

(ii) Let f(x) = (-x)–1 = 1/-x = -1/x . Accordingly, f(x + h) = -1/(x + h) 
By first principle,  

(iii) Let f(x) = sin (x + 1). Accordingly, f(x + h) = sin(x + h+ 1) 
By first principle,  

= cos(x + 1) 

(iv) Let f(x) = cos (x - π/8). Accordingly, f(x + h) = cos (x + h - π/8) 
By first principle,  


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, 


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,  

= (ax + b)(2c2 x + 2cd) + (cx + d2)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,  

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  

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,  

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,  

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,  

= - cosec2 x 
∴ (cot x)' = -cosec2 x ...(2) 
Now, let f2 (x) = cosec x . Accordingly, f2 (x + h) = cosec (x + h) 
By first principle,  

= - cosec x . cot x 
∴ (cosec x)' = -cosec x. cot x  ...(3) 
From (1), (2), and (3), we obtain 
f '(x) = cosec x(- cosec2 x) + cot x( -cosec x cot x) 
= -cosec3 x - cot2 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,  

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,  

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) 

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 = sin3 x . 
∴ dy/dx = (d/dx)(sin x sin2 x) 
= (sin x)' sin2 x + sin x(sin2 x)'  [By Leibnitz product rule] 
= cos x sin2 x + 2 sin2 x cos x  
= 3 sin2 x cos x 

= (sin x)' sink x + sin x(sink x)'   [By Leibnitz product rule]
= cos x sink x + sin x(k sin(k-1) x cos x)   [Using (2)] 
= cos x sink x  + k sink x cos x
= (k + 1)sink 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,  

Let g(x) = sin (x + a). Accordingly, g(x + h) = sin (x + h + a) 
By first principle,  

= cos (x + a)  ...(ii) 
From (i) and (ii), we obtain  

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,  

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) = (x2 + 1) cos x
By product rule,  

= (x2 + 1)(- sin x) + cos x(2x) 
= -x2 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) = (ax2 + sin x)(p + q cos x)
By product rule, 

= (ax2 + sin x)(-q sin x)  + (p + q cos x)(2ax + cos x) 
= -q sin x (ax2 + 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,  

Let g(x) = tan x. Accordingly, g(x + h) = tan (x + h) 
By first principle,  

= sec2 x  ...(ii) 
Therefore, from (i) and (ii), we obtain  
f '(x) = (x + cos x)(1 - sec2 x) + (x - tan x)(1 - sin x) 
= (x + cos x)(-tan2 x) + (x - tan x)(1 - sin x) 
= -tan2 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

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,  

Let f1 (x) = tan x, f2 (x)  = sec x 
Accordingly, f1 (x + h) = tan (x + h) and f2(x + h) = sec (x + h) 

⇒ (d/dx) sec x = sec x tan x ...(iii) 
From (i), (ii) and (iii), we obtain  
f'(x) = (x + sec x) (1 - sec2 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/sinn x  
By quotient rule,  

Previous Post Next Post