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

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

Differentiation

This content was designed by the ‘Basics in Maths‘ team. These notes to do help 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} (ax + b)  

            = n (ax + b) ^{n – 1} a

            = an (ax + b) ^{n –} 1

(ii)  f(x) = x² 2^x log x

Sol:

Given f(x) = x² 2^x log x

f’ (x) = (x²) 2^x log x + x² (2^x) log x + x² 2^x (log x).

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

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

(iii)  f(x) = (x > 0)

Sol:

Given f(x) =

f’ (x) =   . log 7    (x³ + 3x)

          =     log 7 (3x² + 3)

          =3 (x² + 1)  log 7

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

Sol:

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

f’ (x) = (sec x + tan x)

         = (sec² x + sec x tan x)

         =  sec x (sec x + tan x)

        = sec x

Question 2

Find the derivative of  the following  functions

(i) f(x) = e^x (x² + 1)

Sol:

Given f(x) = e^x (x² + 1)

 f’ (x) = e^x (x² + 1) + (x² + 1)  (e^x)

          = e^x (2x + 0) + (x² + 1) e^x

            = e^x (x² + 2x + 1)

            = e^x (x + 1)²

(ii)

 Let y =

(iii) cos (log x + e^x) 

(iv) x = tan (e^-y)

e^-y = tan^-1 x

(v) cos [log (cot x)]

(vi) sin[tan^-1(e^x)]

(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

 = 3  (cos^-1 x)

      = 3

     =

(viii)

(ix)  

(x)

Differentiation

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 (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

                   =

                   = 50 × 101

                  = 5050

Question 5

 From the following functions. Find their derivatives.

Question 6 

If y = , find

Sol:

Given y =

Question 7

If y = log (cosh 2x), find

Sol:

Given y = log (cosh 2x)

Question 8

If x = a cos³ t, y = a sin³ t, find

Sol:

Given If x = a cos³ t, y = a sin³ t

Question 9

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

(i) f(x) = e^x, g(x) =

f’ (x) = e^x and g’ (x) =

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

(ii )   

  put x = tan θ ⟹ θ = tan^-1 x

Question 10

if y = then prove that

Sol:

Given y =

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