Q. 7

# Differentiate the following functions:

(i) (2x + 3) (3x – 5)

(ii) x(1 + x)^{3}

(iii)

(iv)

(v)

(vi) (2x^{2} + 5x – 1) (x – 3)

Answer :

Formula:

Chain rule -

Where u and v are the functions of x.

(i) (2x + 3) (3x – 5)

Applying, Chain rule

Here, u = 2x + 3

V = 3x -5

(2x + 3) (3x – 5)

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

= 6x + 9 + 6x -10

= 12x -1

(ii) x(1 + x)^{3}

Applying, Chain rule

Here, u = x

V = (1 + x)^{3}

x(1 + x)^{3}

= x×3×(1 + x)^{2} + (1 + x)^{3}(1)

= (1 + x)^{2}(3x+x+1)

= (1 + x)^{2}(4x+1)

(iii) = (x^{1/2} + x^{-1})(x – x^{-1/2} )

Applying, Chain rule

Here, u = (x^{1/2} + x^{-1})

V = (x – x^{-1/2} )

(x^{1/2} + x^{-1})(x – x^{-1/2} )

= (x^{1/2} + x^{-1})(x – x^{-1/2} ) + (x – x^{-1/2} )(x^{1/2} + x^{-1})

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

= x^{1/2} + x^{-1} + x^{-1} + x^{-5/2} + x^{1/2} – x^{-1} - x^{-1} + x^{-5/2}

= x^{1/2} + x^{-5/2}

(iv)

Differentiation of composite function can be done by

Here, f(g) = g^{2} , g(x) =

= 2g×(1 + )

= 2( (1 + )

= 2(x + - + )

= 2(x + )

(v)

Differentiation of composite function can be done by

Here, f(g) = g^{3} , g(x) = x^{2} -

= 3g^{2}×(2x - )

= 3 (2x - )

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

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

(vi) (2x^{2} + 5x – 1) (x – 3)

Applying, Chain rule

Here, u = (2x^{2} + 5x – 1)

V = (x – 3)

(2x^{2} + 5x – 1) (x – 3)

(2x^{2} + 5x – 1)(2x^{2} + 5x – 1)

= (2x^{2} + 5x – 1)×1 + (x – 3)(4x + 5)

= 2x^{2} + 5x – 1 + 4x^{2} -7x -15

= 6x^{2} -2x -16

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