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Limits And Derivatives

Class 11th Mathematics CBSE Solution
Exercise 13.1
  1. lim_ x arrow3 x+3 Evaluate the following limits in Exercises.
  2. lim_ x arrow pi (x- 22/7) Evaluate the following limits in Exercises.…
  3. lim_ r arrow1 pi r^2 Evaluate the following limits in Exercises.
  4. lim_ x arrow4 4x+3/x-2 Evaluate the following limits in Exercises.…
  5. lim_ x arrow-1 x^10 + x^5 + 1/x-1 Evaluate the following limits in Exercises.…
  6. Evaluate the following limits in Exercise. lim_ x arrow0 (x+1)^5 -1/x…
  7. lim_ x arrow2 3x^2 - x-10/x^2 - 4 Evaluate the following limits in Exercises.…
  8. Evaluate the following limits in Exercises. lim_ x arrow3 x^4 - 81/2x^2 - 5x-3…
  9. lim_ x arrow0 ax+b/cx+1 Evaluate the following limits in Exercises.…
  10. lim_ z arrow1 z^1/3 - 1/z^1/b - 1 Evaluate the following limits in Exercises.…
  11. lim_ z arrow1 ax^2 + bx+c/cx^2 + bx+a , a+b+c not equal 0 Evaluate the…
  12. lim_ x arrow-2 1/x + 1/2/x+2 Evaluate the following limits in Exercises.…
  13. lim_ x arrow0 sinax/bx Evaluate the following limits in Exercises.…
  14. lim_ x arrow0 sinax/sinbx , a , b not equal 0 Evaluate the following limits in…
  15. lim_ x arrow pi sin (pi -x)/pi (pi -x) Evaluate the following limits in…
  16. lim_ x arrow0 cosx/pi -x Evaluate the following limits in Exercises.…
  17. lim_ x arrow0 cos2x-1/cosx-1 Evaluate the following limits in Exercises.…
  18. lim_ x arrow0 ax+xcosx/bsinx Evaluate the following limits in Exercises.…
  19. lim_ x arrow0 xsecx Evaluate the following limits in Exercises.
  20. lim_ x arrow0 sinax+bx/ax+sinbxa , b , a+b not equal 0 Evaluate the following…
  21. lim_ x arrow0 (cosex-cotx) Evaluate the following limits in Exercises.…
  22. lim_ x arrow pi /2 tan2x/x - pi /2 Evaluate the following limits in Exercises.…
  23. Find lim_ x arrow0 f (x) and lim_ x arrow1 f (x) , where f (x) = 2x+3x less…
  24. Find lim_ x arrow1 f (x) , where f (x) = c x^2 - 1x less than equal to 1 - x^2…
  25. Evaluate lim_ x arrow0 f (x) , where f(x) = |x|/x , x not equal 0 x 0 , x = 0…
  26. Find lim_ x arrow0 f (x) , where f (x) = x/|x| , x not equal 0 0 ,…
  27. Find lim_ x arrow5 f (x) , where f (x) = |x|-5
  28. Suppose f (x) = cc a+bx , 1 4 , & x = 1 b-ax1 and if lim_ x arrow1 f (x) = f…
  29. Let a1, a2,… , an be fixed real numbers and define a function f (x) = (x − a1)…
  30. If f (x) = c |x|+1 , x0 0 , x = 0 |x|-1 , x0 For what value (s) of a does lim_…
  31. If the function f(x) satisfies lim_ x arrow1 f (x) - 2/x^2 - 1 = pi , evaluate…
  32. If f (x) = ll mx^2 + n , 0 nx+m , & 0 less than equal to x less than equal to…
Exercise 13.2
  1. Find the derivative of x^2 - 2 at x = 10.
  2. Find the derivative of 99x at x = l00.
  3. Find the derivative of x at x = 1.
  4. x^3 - 27 Find the derivative of the following functions from first principle.…
  5. (x −1) (x − 2) Find the derivative of the following functions from first…
  6. 1/x^2 Find the derivative of the following functions from first principle.…
  7. x+1/x-1 Find the derivative of the following functions from first principle.…
  8. For the function f (x) = x^100/100 + x^99/99+... x^2/2+x+1 .Prove that f’ (1)…
  9. Find the derivative of x^n + ax^n-1 + a^2x^n-2 + l +a^n-1x+a^n for some fixed…
  10. (x − a) (x − b) For some constants a and b, find the derivative of…
  11. (ax^2 + b)^2 For some constants a and b, find the derivative of
  12. x-a/x-b For some constants a and b, find the derivative of
  13. Find the derivative of x^n - a^n/x-a for some constant a.
  14. 2x - 3/4 Find the derivative of
  15. (5x^3 + 3x-1) (x-1) Find the derivative of
  16. x^-3 (5+3x) Find the derivative of
  17. x^3 (3-6x^-8) Find the derivative of
  18. x^-4 (3-4x^-3) Find the derivative of
  19. 2/x+1 - x^2/3x-1 Find the derivative of
  20. Find the derivative of cos x from first principle.
  21. sin x cos x Find the derivative of the following functions:
  22. sec x Find the derivative of the following functions:
  23. 5sec x + 4cos x Find the derivative of the following functions:
  24. cosec x Find the derivative of the following functions:
  25. 3 cot x + 5 cosec x Find the derivative of the following functions:…
  26. 5 sin x - 6 cos x + 7 Find the derivative of the following functions:…
  27. 2tan x - 7sec x Find the derivative of the following functions:
Miscellaneous Exercise
  1. -x Find the derivative of the following functions from first principle:…
  2. (-x)-1 Find the derivative of the following functions from first principle:…
  3. sin (x + 1) Find the derivative of the following functions from first…
  4. cos (x - pi /8) Find the derivative of the following functions from first…
  5. Find the derivative of the following functions (it is to be understood that a,…
  6. Find the derivative of the following functions (it is to be understood that a,…
  7. (ax+b) (cx+d)^2 Find the derivative of the following functions (it is to be…
  8. ax+b/cx+d Find the derivative of the following functions (it is to be…
  9. 1 + 1/x/1 - 1/x Find the derivative of the following functions (it is to be…
  10. 1/ax^2 + bx+c Find the derivative of the following functions (it is to be…
  11. ax+b/px^2 + qx+r Find the derivative of the following functions (it is to be…
  12. px^2 + qx+r/ax+b Find the derivative of the following functions (it is to be…
  13. a/x^4 - b/x^2 + cosx Find the derivative of the following functions (it is to…
  14. 4 root x-2 Find the derivative of the following functions (it is to be…
  15. (ax + b)n Find the derivative of the following functions (it is to be…
  16. (ax+b)^n (cx+d)^m Find the derivative of the following functions (it is to be…
  17. sin (x+a) Find the derivative of the following functions (it is to be…
  18. cosec x cot x Find the derivative of the following functions (it is to be…
  19. cosx/1+sinx Find the derivative of the following functions (it is to be…
  20. sinx+cosx/sinx-cosx Find the derivative of the following functions (it is to…
  21. secx-1/secx+1 Find the derivative of the following functions (it is to be…
  22. Find the derivative of the following functions (it is to be understood that a,…
  23. a+bsinx/c+dcosx Find the derivative of the following functions (it is to be…
  24. sin (x+a)/cosx Find the derivative of the following functions (it is to be…
  25. x^4 (5sinx-3cosx) Find the derivative of the following functions (it is to be…
  26. (x^3 + 1) cosx Find the derivative of the following functions (it is to be…
  27. (ax^2 + sinx) (p+acosx) Find the derivative of the following functions (it is…
  28. (x+cnsx) (x-tanx) Find the derivative of the following functions (it is to be…
  29. 4x+5sinx/3x+7cosx Find the derivative of the following functions (it is to be…
  30. x^2cos (pi /4)/sinx Find the derivative of the following functions (it is to…
  31. x/1+tanx Find the derivative of the following functions (it is to be…
  32. (x+secx) (x-tanx) Find the derivative of the following functions (it is to be…
  33. x/sin^nx Find the derivative of the following functions (it is to be…
Exercise 13.1
Question 1.

Evaluate the following limits in Exercises.
Answer:

⇒

= 3 + 3


= 6
Question 2.

Evaluate the following limits in Exercises.
Answer:

⇒



Question 3.

Evaluate the following limits in Exercises.
Answer:

⇒ =π (1)2 = π
Question 4.

Evaluate the following limits in Exercises.
Answer:

⇒



Question 5.

Evaluate the following limits in Exercises.
Answer:

⇒


=


=
Question 6.

Evaluate the following limits in Exercise.
Answer:

⇒

As this limit is undefined.


Let x + 1 = y.


So, x = y – 1


⇒


We know that =nan-1


⇒


=5(1)5-1

= 5(1)4

= 5
Question 7.

Evaluate the following limits in Exercises.
Answer:


Evaluating the limit at x = 2 we get,
⇒

Now we need to factorize the numerator, So,

Now putting the value x = 2 we get,
Question 8.

Evaluate the following limits in Exercises.
Answer:

To Find:



First put x = 3 in the limit,



As the limit is of the form 0/0 we need to factorize numerator and denominator,
Question 9.

Evaluate the following limits in Exercises.
Answer:

⇒



Question 10.

Evaluate the following limits in Exercises.
Answer:

⇒


Let z1/6 = x



⇒


We know that =nan-1


⇒


Question 11.

Evaluate the following limits in Exercises.
Answer:

⇒ [∵ a + b + c ≠ 0]
Question 12.

Evaluate the following limits in Exercises.
Answer:

⇒


So,



Question 13.

Evaluate the following limits in Exercises.
Answer:


Formula to be used:



Applying the limits in the given expression we get,

⇒


Multiplying and dividing the given expression by a we get,




We know that:



Question 14.

Evaluate the following limits in Exercises.
Answer:

⇒


So,


⇒


We know that




Question 15.

Evaluate the following limits in Exercises.
Answer:

⇒


We know that


⇒


Ans.



Question 16.

Evaluate the following limits in Exercises.
Answer:

⇒


Ans.



Question 17.

Evaluate the following limits in Exercises.
Answer:

⇒


So,


[∵ cos2x = 1 – 2sin2x]


⇒


⇒ 4


⇒ 4


We know that


⇒ 4× 12/12


⇒ 4


Ans.



Question 18.

Evaluate the following limits in Exercises.
Answer:

⇒


So,


=


We know that


=


Ans.



Question 19.

Evaluate the following limits in Exercises.
Answer:

⇒


Ans.



Question 20.

Evaluate the following limits in Exercises.
Answer:

⇒


So,


=


We know that


=


=


=1


Ans.


Question 21.

Evaluate the following limits in Exercises.
Answer:



Applying the formulas,




Now, applying the formula,




Hence, the answer is 0.
Question 22.

Evaluate the following limits in Exercises.
Answer:

⇒


Let x – (π/2) = y


So, x→ (π/2) = y→ 0


Now,


=


=


We know that tan x = sin x/ cos x


=


Multiply and divide by 2,


=


=


=


=


=2


Ans.



Question 23.

Find and , where


Answer:

Given function is


:


⇒


⇒


∴


Now :


⇒


⇒


∴


Ans. =3 and =6
Question 24.

Find , where
Answer:

Given function is


:


⇒


⇒


Here


∴ does not exist.
Question 25.

Evaluate , where f(x) =
Answer:

Given function is f(x) =



We know that,


exists only when



Here we need to prove that:


As we also know,

|x| = x , if x>0
= -x,if x<0

So,


⇒


⇒


Here


∴ does not exist.
Question 26.

Find, where
Answer:

Given function is


:


⇒


⇒


Here


∴ does not exist.
Question 27.

Find , where


Answer:

Given function is f(x) =


:


⇒


⇒


∴


Ans.



Question 28.

Suppose

and if = f (1) what are possible values of a and b?
Answer:

Given function is f(x) =


And


⇒


⇒


Here, f(1) = 4


∴


So, a + b = 4 and b – a = 4


Solving the above two equations, we get


⇒ a = 0 and b = 4


Ans. The possible values of a and b for the given function f(x) are 0 and 4 respectively.
Question 29.

Let a1, a2,… , an be fixed real numbers and define a function

f (x) = ( x − a1) ( x − a2)...( x − an) .

What is ? For some compute .
Answer:

Given function is f(x) = (x – a1) (x – a2) … (x – an)


:


⇒


=


= (a1 – a1) (a1 – a2) … (a1 – an) = 0


∴


:


⇒


=


= (a – a1) (a – a2) … (a – an)


∴


Ans. and



Question 30.

If For what value (s) of a does exists?
Answer:

Given function is


There are three cases.


Case 1: When a = 0


:


⇒


⇒


Here


∴ does not exist.


Case 2: When a < 0


:


⇒


⇒


∴


∴ lim(f(x)) exists at x = a when a < 0


Case 3: When a > 0


:


⇒


⇒


∴


∴ lim(f(x)) exists at x = a when a > 0


Ans. exists for all a ≠ 0
Question 31.

If the function f(x) satisfies , evaluate
Answer:

Given that function f(x) satisfies


⇒


⇒


⇒


⇒


⇒


⇒


⇒


⇒


Ans.



Question 32.

If For what integers m and n does both and exist?
Answer:

Given function is


:


⇒


⇒


∴ exists if n = m.


Now :


⇒


⇒


∴


∴ exists for any integral value of m and n.
Exercise 13.2
Question 1.

Find the derivative of x2 – 2 at x = 10.
Answer:

Let f(x) = x2 – 2,


From first principle






= 20 + 0 = 20


Thus, the derivative of x2 – 2 at x = 10 is 20.
Question 2.

Find the derivative of 99x at x = l00.
Answer:

Let f(x) = 99x,


From first principle






= 99


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,


From first principle






= 1


Thus, the derivative of x at x = 1 is 1.
Question 4.

Find the derivative of the following functions from first principle.

x3 – 27
Answer:

Let f(x) = x3 – 27


Accordingly, from first principle





= 0 + 3x2 = 3x2
Question 5.

Find the derivative of the following functions from first principle.

(x −1) (x − 2)
Answer:

Let f(x) = (x – 1)(x – 2)


Accordingly, from first principle





= 0 + 2x – 3 = 2x – 3
Question 6.

Find the derivative of the following functions from first principle.
Answer:

Let


Accordingly, from the first principle,
Question 7.

Find the derivative of the following functions from first principle.
Answer:

Let


From first principle,
Question 8.

For the function .Prove that f’ (1) =100 f’ (0).
Answer:

Given function is





Since,




At x = 0


f’(0) = 0 + 0 + … + 0 + 1


f’(0) = 1


At x = 1


f’(1) = 199 + 198 + … + 1 + 1 = [1 + 1 ….+ 1] 100 times = 1 × 100 = 100


Thus f’(1) = 100 f’(0)
Question 9.

Find the derivative of for some fixed real number a.
Answer:

Given f(x) =




Since,


f’(x) = nxn-1 + a(n-1)xn-2 + a2(n – 2)xn-3 + … + an-1 + an(0)


f’(x) = nxn-1 + a(n-1)xn-2 + a2(n – 2)xn-3 + … + an-1
Question 10.

For some constants a and b, find the derivative of

(x − a) (x − b)
Answer:

Let f(x) = (x – a)(x – b)


⇒ f(x) = x2 – (a + b)x + ab




Since,


f’(x) = 2x – (a + b) + 0


= 2x – a – b


∴ f’(x) = 2x – a – b
Question 11.

For some constants a and b, find the derivative of
Answer:

Let f(x) =


⇒ f(x) = a2x4 + 2abx2 + b2


So,


⇒


Since,


f’(x) = a2 × 4x3 + 2ab × 2x + 0


= 4a2x3 + 4abx


∴ f’(x) = 4ax (ax2 + b)
Question 12.

For some constants a and b, find the derivative of
Answer:

Let


By division rule



=


=




Question 13.

Find the derivative of for some constant a.
Answer:

Let


⇒


By quotient rule,




∴



Question 14.

Find the derivative of
Answer:

Let f(x) =





∴ f’(x) = 2
Question 15.

Find the derivative of
Answer:

Let f(x) =


By product rule,




= (5x3 + 3x – 1) + (x – 1)(15x2 + 3)


= 5x3 + 3x – 1 + 15x3 + 3x – 15x2 – 3


= 20x3 – 15x2 + 6x – 4
Question 16.

Find the derivative of
Answer:

Let f(x) = x-3 (5 + 3x)


By product rule,
Question 17.

Find the derivative of
Answer:

Let f(x) =


By product rule,



= x5 (0 – 6 × (-9) x-9-1) + (3 – 6x-9)(5x4)


= x5 (54x-10) + 15x4 – 30x-5


∴ f’(x) = 24x-5 + 15x4
Question 18.

Find the derivative of
Answer:

Let f(x) =


By product rule,





= x-4 (20x-6) + (3 – 4x-5)(-4x-5)


= 20x-10 – 12x-5 + 16x-10


f’(x) = 36x-10 – 12x-5
Question 19.

Find the derivative of
Answer:

Let f(x) =



By quotient rule,
Question 20.

Find the derivative of cos x from first principle.
Answer:

Let f(x) = cos x. Accordingly, f(x + h) = cos (x + h)


By first principle,







= -sin(x)
Question 21.

Find the derivative of the following functions:

sin x cos x
Answer:

Let f(x) = sin x cos x


By product rule,



=sinx × (-sinx) + cosx × cosx


= - sin2 x + cos2 x


∴ f’(x) = cos2 x – sin2 x
Question 22.

Find the derivative of the following functions:

sec x
Answer:

Let f(x) = sec x = 1/cos x



By quotient rule,





∴ f’(x) = tan x sec x
Question 23.

Find the derivative of the following functions:

5sec x + 4cos x
Answer:

Let f(x) = 5sec x + 4cos x





∴ f’(x) = 5 sec x tan x – 4 sin x
Question 24.

Find the derivative of the following functions:

cosec x
Answer:

Let f(x) = cosec x, accordingly f(x + h) = cosec (x + h)


By first principle



=


=


=


=


=


∴ f’(x) = -cosec x cot x
Question 25.

Find the derivative of the following functions:

3 cot x + 5 cosec x
Answer:

Let f(x) = 3 cot x + 5 cosec x


f’(x) = 3 (cot x)’ + 5 (cosec x)’


Let f1(x) = cot x, accordingly f1(x + h) = cot (x + h)


By first principle



=


=


=


=


=


∴ (cot x)’ = - cosec2 x ………………..(2)


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


So, f’(x) = 3 (cot x)’ + 5 (cosec x)’


Putting (cot x)’ and (cosec x)’ in f’(x)


f’(x) = 3 × (-cosec2 x) + 5 × (-cosec x cot x)


f’(x) = -3cosec2 x – 5cosec x cot x
Question 26.

Find the derivative of the following functions:

5 sin x – 6 cos x + 7
Answer:

Let f(x) = 5 sin x – 6 cos x + 7




= 5 × cos x – 6 × (- sin x) + 0


∴ f’(x) = 5cos x + 6sin x
Question 27.

Find the derivative of the following functions:

2tan x – 7sec x
Answer:

Let f(x) = 2 tan x – 7 sec x




f’(x) = 2 × (sec2 x) – 7 × (sec x tan x)


f’(x) = 2sec2 x – 7sec x tan x
Miscellaneous Exercise
Question 1.

Find the derivative of the following functions from first principle:

–x
Answer:

Let f(x) = –x. Accordingly, f(x + h) = -(x + h)


By first principle,






= 1
Question 2.

Find the derivative of the following functions from first principle:

(-x)-1
Answer:

Let f(x) = (–x)-1 = -1/x. Accordingly,


By first principle,



=


=


=


= 1/x2
Question 3.

Find the derivative of the following functions from first principle:

sin (x + 1)
Answer:

Let f(x) = sin (x + 1) . Accordingly, f(x + h) = sin (x + h + 1)


By first principle,







= cos (x + 1)
Question 4.

Find the derivative of the following functions from first principle:
Answer:

Let f(x) = . Accordingly,


By first principle,




=



=


=



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): (x + a)
Answer:

Let f(x) = x + a. Accordingly, f(x + h) = x + h + a


By first principle,



=


=


=


= 1
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):
Answer:

Let









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):
Answer:

Let f(x) = (ax + b)(cx + d)2


By product rule,
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):
Answer:

Let


By division rule



=


=




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):
Answer:

Let


By quotient rule



=


=


=


∴



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):
Answer:

Let


By quotient rule



=


=




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):
Answer:

Let


By quotient rule



=


=




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):
Answer:

Let


By quotient rule



=


=




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):
Answer:

Let



=


=


=


∴



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):
Answer:

Let f(x) =



=


=


∴



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):

(ax + b)n
Answer:

Let f(x) = (ax + b)n


By first principle



=


=


=


∴ f’(x) = na (ax + b)n-1
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):
Answer:

Let f(x) =


By product rule





f’(x) = na (cx + d)m (ax + b)n-1 + mc (ax + b)n (cx + d)m-1


∴ f’(x) = (ax + b)n-1 (cx + d)m-1 [na (cx + d) + mc (ax + b)]
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):

sin (x+a)
Answer:

Let f(x) = sin (x + a), accordingly f(x + h) = sin (x + h + a)


By first principle,



=


=


=


=


= cos (x + a)
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):

cosec x cot x
Answer:

Let f(x) = cosec x cot x


By product rule


f’(x) = cosec x (cot x)’ + cot x (cosec x)’ ……..(1)


Let f1(x) = cot x, accordingly f1(x + h) = cot (x + h)


By first principle



=


=


=


=


=


∴ (cot x)’ = - cosec2 x ………………..(2)


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)


Putting the value of (2) and (3) in (1)


f’(x) = cosec x (-cosec2 x) + cot x (-cosec x cot x)


= - cosec3 x – cot2 x cosec x
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):
Answer:

Let f(x) =


By quotient rule:
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):
Answer:

Let f(x) =


By quotient rule,








[we know that, sin2x + cos2x = 1]
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):
Answer:

Let f(x) =



By quotient rule,
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):


Answer:

Let f(x) =







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):
Answer:

Let f(x) =


By quotient rule,
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):
Answer:

Let f(x) =


By quotient rule,
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):
Answer:

Let f(x) =


By product rule,
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):
Answer:

Let f(x) =


By product rule



=(x2 + 1)(-sinx) + cosx × 2x


= -x2sinx - sinx+2x cosx


∴ f’(x) = - x2 sin x – sin x + 2x cos x
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):
Answer:

Let f(x) =


By product rule,




∴ f’(x) = (ax2 + sin x) (-q sin x) + (p + q cos x) (2ax + cos x)
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):
Answer:

Let f(x) =


By product rule,




= (x + cos x) (1 – sec2 x) + (x – tan x) (1 – sin x)


∴ f’(x) = - tan2 x (x + cos x) + (x – tan x) (1 – sin x)
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):
Answer:

Let f(x) =


By quotient rule,
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):
Answer:

Let f(x) =


By quotient rule,
Question 31.

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):
Answer:

Let f(x) =


By quotient rule,
Question 32.

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):
Answer:

Let f(x) =


By product rule,
Question 33.

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):
Answer:

Let f(x) =


By quotient rule,