Question

For some constants a and b , find the derivative of (i) ( x – a ) ( x – b ) (ii) ( ax 2 + b ) 2 (iii)

Answer 1

(I)

Let the given function be:

f( x )=( x−a )( x−b )

The derivative of the given function.

On expanding the function:

f( x )=( x 2 −ax−bx−ab )

The formula for derivative is:

d( x n ) dx =( n x n−1 )

Also f ' ( x )= d dx f( x )

On applying the formula of derivatives from above:

d dx f( x )= d dx ( x 2 −ax−bx+ab ) = d dx ( x 2 )−a d dx ( x )−b d dx ( x )+ d dx ( ab ) =2x−a−b

Thus, the derivative of the function f( x )=( x 2 −ax−bx−ab ) is 2x−a−b .

(II)

Let the given function be:

f( x )= ( a x 2 +b ) 2

The formula of derivatives is

d( x n ) dx =( n x n−1 )

On applying the formula, we get

d dx f( x )= d dx ( a x 2 +b ) 2 = d dx ( a 2 x 4 + b 2 +2ab x 2 ) = d dx a 2 x 4 + d dx ( b 2 )+2ab d dx ( x 2 ) =4 a 2 x 3 +0+2ab( 2x )

On further simplification, we get

f ' ( x )=4 x 3 a 2 +4abx =4ax( x 2 a+b )

Thus, the derivative of the given function f( x )= ( a x 2 +b ) 2 is 4ax( x 2 a+b ) .

(III)

Let the given function be:

f( x )= x−a x−b

The formula of derivatives is:

d( x n ) dx =( n x n−1 )

And from quotient formula we know that:

d dx ( U V )= ( UV'−VU' ) V 2

Where, U and V are the derivatives of their respective functions.

In the above question, U=( x−a ) and V=( x−b )

On applying formula:

d dx ( x−a x−b )= ( x−b )f'( x−a )−( x−a )f'( x−b ) ( x−b ) 2 = ( x−b )−( x−a ) ( x−b ) 2 = x−x−b+a ( x−b ) 2 = a−b ( x−b ) 2

Thus, the derivative of the given function f( x )= x−a x−b is a−b ( x−b ) 2 .