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

Differentiation

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

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

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