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) = (5x³ + 3x - 1)(x - 1)

By Leibnitz product rule,

f'(x) = (5x³ + 3x - 1) d/dx (x - 1) + (x - 1) d/dx (5x³ + 3x - 1)

= (5x³ + 3x - 1)(1) + ( x - 1)(5.3x² + 3 - 0)

= (5x³ + 3x - 1) + (x - 1)(15x² + 3)

= 5x³ + 3x - 1 + 15x³ + 3x - 15x² - 3

= 20x³ - 15x² + 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/x⁴ (5 + 2x)

(iv) Let f (x) = x⁵ (3 - 6x^{- 9})

By Leibnitz product rule,

f '(x) = x⁵ d/dx (3 - 6x^{- 9}) + (3 - 6x^{- 9}) d/dx (x⁵)

= x⁵ [0 - 6(- 9) x^{- 9 - 1}]+ (3 - 6x^{- 9} )(5x⁴)

= x⁵ (54x^{- 10} ) + 15x⁴ - 30x^{- 5}

= 54x^{- 5} - 30x^{- 5} + 15x⁴

= 24x^{- 5} + 15x⁴

= 15x⁴ + 24/x⁵

(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/x⁵ + 36/x¹⁰

(vi) Let f (x) = 2/(x + 1) - x²/(3x - 1)

f' (x) = d/dx [2/(x + 1) - x²/(3x - 1)]

By quotient rule,

f' (x) = [(x + 1) d/dx (2) - 2 d/dx (x + 1)] / (x + 1)² - [(3x - 1) d/dx (x²)- x² d/dx (3x - 1)] / (3x - 1)²

= [(x + 1)(0) - 2(1)]/(x + 1)² - [(3x - 1)(2x) - x² (3)]/(3x - 1)²

= - 2/(x + 1)² - [6x² - 2x - 3x²]/(3x - 1)²

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

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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) 20x³ - 15x² + 6x - 4 (iii) - 3/x⁴ (5 + 2x) (iv) 15x⁴ + 24/x⁵ (v) -12/x⁵ + 36/x¹⁰ (vi) - 2/(x + 1)² - [x (3x - 2)]/(3x - 1)²