Welcome To Basics In Maths

# 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 (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) = (x2) 2x log x + x2 (2x) log x + x2 2x (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) = (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) = ex (x2 + 1)

Sol:

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

f’ (x) = ex (x2 + 1) + (x2 + 1)  (ex)

= ex (2x + 0) + (x2 + 1) ex

= ex (x2 + 2x + 1)

= ex (x + 1)2

(ii)

Let y =

(iii) cos (log x + ex)

(iv) x = tan (e-y)

e-y = tan-1 x

(v) cos [log (cot x)]

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

(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)

#### 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) = ex, g(x) =

f’ (x) = ex 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 =