Question

Find the derivative of for some constant a .

Answer 1

Consider the given function,

f( x )= ( x n − a n ) ( x−a )

The quotient rule to find the derivative of the function is,

d dx ( U V )= ( U V ′ −V U ′ ) V 2

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

Differentiate the given function with respect to x by quotient rule,

d dx ( x n − a n x−a )= ( x−a ) d dx ( x n − a n )−( x n − a n ) d dx ( x−a ) ( x−a ) 2 = ( x−a )( n x n−1 )−( x n − a n )( 1 ) ( x−a ) 2 = xn x n−1 − x n −an x n−1 + a n ( x−a ) 2 = n x n − x n −an x n−1 + a n ( x−a ) 2

Thus, the derivative of the given function is n x n −an x n−1 − x n + a n ( x−a ) 2 .