Using mathematical induction prove that for all positive integers n .

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Solution

Let, P( n ): d dx ( x n )=n x n−1 for all positive integers.

For n=1,

P( 1 ): d dx ( x )=1× x 1−1

Thus, the value of P( n )is true for n=1.

Let the value of P( k )be true for some positive integer k.

P( k ): d dx ( x ) k =k x k−1

Consider the value of P( k+1 )is true for some positive integer ( k+1 ).

d dx ( x k+1 )= d dx ( x× x k ) = x k d dx ( x )+x d dx ( x k ) = x k ×1+x×k x k−1 = x k ( k+1 )

Further simplify,

d dx ( x k+1 )=( k+1 ) x ( k+1 )−1

Thus, the value of P( k+1 )is true if P( k )is true.

Hence, by mathematical induction, the given statement is true for every positive integer.

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