Differentiation(2m Q & S) || V.S.A.Q’S||

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This content designed by the ‘Basics in Maths‘ team. These notes to do help the intermediate First-year Maths students.

Inter Maths – 1B two mark questions and solutions are very useful in IPE examinations.


Differentiation

Question 1 

Find f’ (x) for the following functions

(i) f(x) = (ax + b) (x > -b/a)

Sol:

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

 f’ (x) = n (ax + b) n – 1 Differentiation 1(ax + b)  

            = n (ax + b) n – 1 a

            = an (ax + b) n – 1

(ii)  f(x) = x2 2x log x

Sol:

Given f(x) = x2 2x log x

f’ (x) = Differentiation 1(x2) 2x log x + x2 Differentiation 1(2x) log x + x2 2x Differentiation 1(log x).

          = 2×2x log x +x2 2x log a log x + x2 2x (1/x)

          = x 2x[log x2 + x log x log 2 + 1]

(iii)  f(x) =Differentiation 2 (x > 0)

Sol:

Given f(x) =Differentiation 2

f’ (x) = Differentiation 2  . log 7   Differentiation 1 (x3 + 3x)

          =    Differentiation 2 log 7 (3x2 + 3)

          =3 (x2 + 1) Differentiation 2 log 7

(iv) f(x) = log (sec x + tan x)

Sol:

Given, f(x) = log (sec x + tan x)

f’ (x) = Differentiation 6 Differentiation 1(sec x + tan x)

         = Differentiation 6(sec2 x + sec x tan x)

         = Differentiation 6 sec x (sec x + tan x)

        = sec x

Question 2

Find the derivative of  the following  functions

(i) f(x) = ex (x2 + 1)

Sol:

Given f(x) = ex (x2 + 1)

 f’ (x) = ex Differentiation 1(x2 + 1) + (x2 + 1) Differentiation 1 (ex)

          = ex (2x + 0) + (x2 + 1) ex

            = ex (x2 + 2x + 1)

            = ex (x + 1)2

(ii) Differentiation 11

 Let y =Differentiation 11

  Differentiation 12

(iii) cos (log x + ex) 

Differentiation 13  

(iv) x = tan (e-y)

e-y = tan-1 x

Differentiation 15 

(v) cos [log (cot x)]

Differentiation 16

(vi) sin[tan-1(ex)]

Differentiation 17

(vii) cos-1(4x3 – 3x)

let y = cos-1(4x3 – 3x)

put x = cos θ ⟹ θ = cos-1 x

y = cos-1(4 cos 3 θ – 3cos θ)

   = cos-1(cos 3θ)

= 3 θ

= 3 cos-1 x

Differentiation 8 = 3 Differentiation 1 (cos-1 x)

      = 3Differentiation 19

     =Differentiation 20

(viii) Differentiation 21

Differentiation 22

(ix)  Differentiation 23

Differentiation 24

(x) Differentiation 25

Differentiation 26

 

Question 3

Find f’ (x), If f(x) = (x3 + 6 x2 + 12x – 13)100.

Sol:

Given f(x) = (x3 + 6 x2 + 12x – 13)100

f’ (x) = 100(x3 + 6 x2 + 12x – 13)99 Differentiation 1(x3 + 6 x2 + 12x – 13)

          = 100(x3 + 6 x2 + 12x – 13)99 (3x2 + 12 x + 12 – 0)

          =100(x3 + 6 x2 + 12x – 13)99 3 (x2 + 4 x + 4)

          = 300 (x + 2)2 (x3 + 6 x2 + 12x – 13)99

Question 4

If f(x) = 1 + x + x2 + x3 + …. + x100, then find f’ (1).

Sol:

Given f(x) = 1 + x + x2 + x3 + …. + x100

           f’(x) = 0 + 1 + 2x + 3 x2 + … 100 x99

           f’(1) =  1 + 2 + 3 + … + 100

                   =Differentiation 3

                   = 50 × 101

                  = 5050

Question 5

 From the following functions. Find their derivatives.

Differentiation 5

Question 6 

If y =Differentiation 7 , findDifferentiation 8

Sol:

Given y =Differentiation 7

Differentiation 9

Question 7

If y = log (cosh 2x), find Differentiation 8

Sol:

Given y = log (cosh 2x)

Differentiation 10

Question 8

If x = a cos3 t, y = a sin3 t, find

Sol:

Given If x = a cos3 t, y = a sin3 t

Differentiation 18

Question 9

Differentiate f(x) with respect to g(x) for the following.

(i) f(x) = ex, g(x) =Differentiation 27

f’ (x) = ex and g’ (x) = Differentiation 28

derivative of f(x) with respect to g(x) =Differentiation 29

Differentiation 30 

(ii )  Differentiation 31 

  put x = tan θ ⟹ θ = tan-1 x

Differentiation 32 

Question 10

if y = Differentiation 33 then prove thatDifferentiation 34

Sol:

Given y =Differentiation 33

Differentiation 35


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