## 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) = x^{2} 2^{x} log x

**Sol:**

Given f(x) = x^{2} 2^{x} log x

f’ (x) = ^{ }(x^{2}) 2^{x} log x + x^{2} (2^{x}) log x + x^{2} 2^{x} (log x).

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

= x 2^{x}[log x^{2} + x log x log 2 + 1]

**Sol:**

f’ (x) = . log 7 ^{ }(x^{3} + 3x)

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

**Sol:**

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

= sec x

**Question 2**

Find the derivative of the following functions

**(i) **f(x) = e^{x} (x^{2} + 1)

**Sol:**

Given f(x) = e^{x} (x^{2} + 1)

f’ (x) = e^{x} (x^{2} + 1) + (x^{2} + 1) ^{ }(e^{x})

= e^{x} (2x + 0) + (x^{2} + 1) e^{x}

= e^{x} (x^{2} + 2x + 1)

= e^{x} (x + 1)^{2}

**(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}(4x^{3} – 3x)

let y = cos^{-1}(4x^{3} – 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**

**Question 3**

Find f’ (x), If f(x) = (x^{3} + 6 x^{2} + 12x – 13)^{100}.

**Sol:**

Given f(x) = (x^{3} + 6 x^{2} + 12x – 13)^{100}

f’ (x) = 100(x^{3} + 6 x^{2} + 12x – 13)^{99} (x^{3} + 6 x^{2} + 12x – 13)

= 100(x^{3} + 6 x^{2} + 12x – 13)^{99} (3x^{2} + 12 x + 12 – 0)

=100(x^{3} + 6 x^{2} + 12x – 13)^{99} 3 (x^{2} + 4 x + 4)

= 300 (x + 2)^{2} (x^{3} + 6 x^{2} + 12x – 13)^{99}

**Question 4**

If f(x) = 1 + x + x^{2} + x^{3} + …. + x^{100}, then find f’ (1).

**Sol:**

Given f(x) = 1 + x + x^{2} + x^{3} + …. + x^{100}

f’(x) = 0 + 1 + 2x + 3 x^{2} + … 100 x^{99}

f’(1) = 1 + 2 + 3 + … + 100

= 50 × 101

= 5050

**Question 5**

From the following functions. Find their derivatives.

**Question 6**

**Sol:**

**Question 7**

**Sol:**

Given y = log (cosh 2x)

**Question 8**

If x = a cos^{3} t, y = a sin^{3} t, find

**Sol:**

Given If x = a cos^{3} t, y = a sin^{3} t

**Question 9**

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

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

put x = tan θ ⟹ θ = tan^{-1} x

**Question 10**

**Sol: **

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