# Find the derivative of (i) 2x - 3/4 (ii) (5x³ + 3x - 1)(x - 1)

(iii) x⁻ ³ (5 + 3x) (iv) x⁵ (3 - 6x⁻ ⁹) (v) x⁻ ⁴ (3 - 4x⁻ ⁵)

(vi) 2/(x + 1) - x²/(3x - 1)

**Solution:**

**(i)** Let f (x) = 2x - 3/4

f' (x) = d/dx (2x - 3/4)

= 2 d/dx x - d/dx (3/4)

= 2 (1) - 0

= 2

**(ii)** Let f (x) = (5x^{3} + 3x - 1)(x - 1)

By Leibnitz product rule,

f'(x) = (5x^{3} + 3x - 1) d/dx (x - 1) + (x - 1) d/dx (5x^{3} + 3x - 1)

= (5x^{3} + 3x - 1)(1) + ( x - 1)(5.3x^{2} + 3 - 0)

= (5x^{3} + 3x - 1) + (x - 1)(15x^{2} + 3)

= 5x^{3} + 3x - 1 + 15x^{3} + 3x - 15x^{2} - 3

= 20x^{3} - 15x^{2} + 6x - 4

**(iii)** Let f (x) = x^{- 3} (5 + 3x)

By Leibnitz product rule,

f' ( x) = x^{- }^{3 }d/dx (5 + 3x) + (5 + 3x) d/dx (x^{- }^{3})

= x^{- }^{3} (0 + 3) + (5 + 3x)(- 3x^{- }^{3 }^{- }^{1} )

= x^{- }^{3} (3) + (5 + 3x)(- 3x ^{-}^{4})

= 3x^{- }^{3} - 15x^{- }^{4} - 9x^{- }^{3}

= - 6x^{- 3} - 15x^{- 4}

= - 3x^{- 3} (2 + 5/x)

= - 3x^{- 3}/x (2x + 5)

= - 3/x^{4} (5 + 2x)

**(iv)** Let f (x) = x^{5} (3 - 6x^{- 9})

By Leibnitz product rule,

f '(x) = x^{5 }d/dx (3 - 6x^{- }^{9}) + (3 - 6x^{- }^{9}) d/dx (x^{5})

= x^{5} [0 - 6(- 9) x^{- }^{9 }^{- }^{1}]+ (3 - 6x^{- }^{9} )(5x^{4})

= x^{5} (54x^{- }^{10} ) + 15x^{4} - 30x^{- }^{5}

= 54x^{- 5} - 30x^{- 5} + 15x^{4}

= 24x^{- 5} + 15x^{4}

= 15x^{4} + 24/x^{5}

**(v)** Let f (x) = x^{- 4} (3 - 4x - x^{- 5})

By Leibnitz product rule,

f '(x) = x^{-}^{4 }d/dx (3 - 4x^{- }^{5}) + (3 - 4x^{- }^{5}) d/dx (x^{-}^{4})

= x^{- }^{4} [0 - 4 (- 5) x^{-}^{5 }^{- }^{1]} + (3 - 4x^{- }^{5} )(- 4) x^{- }^{4 }^{- }^{1}

= x^{- }^{4} (20x^{- }^{6} ) + (3 - 4x^{- }^{5} )(- 4x^{- }^{5})

= 20x^{- }^{10} - 12x^{- }^{5} + 16x^{- }^{10}

= 36x^{- 10} - 12x^{- 5}

= -12/x^{5} + 36/x^{10}

**(vi)** Let f (x) = 2/(x + 1) - x^{2}/(3x - 1)

f' (x) = d/dx [2/(x + 1) - x^{2}/(3x - 1)]

By quotient rule,

f' (x) = [(x + 1) d/dx (2) - 2 d/dx (x + 1)] / (x + 1)^{2} - [(3x - 1) d/dx (x^{2})- x^{2} d/dx (3x - 1)] / (3x - 1)^{2}

= [(x + 1)(0) - 2(1)]/(x + 1)^{2} - [(3x - 1)(2x) - x^{2} (3)]/(3x - 1)^{2}

= - 2/(x + 1)^{2} - [6x^{2} - 2x - 3x^{2}]/(3x - 1)^{2}

= - 2/(x + 1)^{2} - [x (3x - 2)]/(3x - 1)^{2}

NCERT Solutions Class 11 Maths Chapter 13 Exercise 13.2 Question 9

## Find the derivative of (i) 2x - 3/4 (ii) (5x³ + 3x - 1)(x - 1) (iii) x⁻ ³ (5 + 3x) (iv) x⁵ (3 - 6x⁻ ⁹) (v) x⁻ ⁴ (3 - 4x⁻ ⁵) (vi) 2/(x + 1) - x²/(3x - 1)

**Summary:**

The derivatives are (i) 2 (ii) 20x^{3} - 15x^{2} + 6x - 4 (iii) - 3/x^{4} (5 + 2x) (iv) 15x^{4} + 24/x^{5} (v) -12/x^{5} + 36/x^{10} (vi) - 2/(x + 1)^{2} - [x (3x - 2)]/(3x - 1)^{2}

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