Q. 7

# <span lang="EN-US

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)(3x1-1+0) + (3x – 5)(2x1-1+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)

Applying, Chain rule

Here, u = (x1/2 + x-1)

V = (x – x-1/2 )

(x1/2 + x-1)(x – x-1/2 )

= (x1/2 + x-1)(x – x-1/2 ) + (x – x-1/2 )(x1/2 + x-1)

= (x1/2 + x-1)(1+ x-3/2) + (x – x-1/2 )(x-1/2 – x-2)

= x1/2 + x-1 + x-1 + x-5/2 + x1/2 – x-1 - x-1 + x-5/2

= x1/2 + x-5/2

(iv)

Differentiation of composite function can be done by

Here, f(g) = g2 , g(x) =

= 2g×(1 + )

= 2( (1 + )

= 2(x + - + )

= 2(x + )

(v)

Differentiation of composite function can be done by

Here, f(g) = g3 , g(x) = x2 -

= 3g2×(2x - )

= 3 (2x - )

= 3(2x3 - - + )

= 3(2x3 - + )

(vi) (2x2 + 5x – 1) (x – 3)

Applying, Chain rule

Here, u = (2x2 + 5x – 1)

V = (x – 3)

(2x2 + 5x – 1) (x – 3)

(2x2 + 5x – 1)(2x2 + 5x – 1)

= (2x2 + 5x – 1)×1 + (x – 3)(4x + 5)

= 2x2 + 5x – 1 + 4x2 -7x -15

= 6x2 -2x -16

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