Question

Find the derivative of for some fixed real number a .

Answer 1

Let the given function be:

f( x )= x n +a x n−1 + a 2 x n−2 +… a n−1 x+ a n

The derivative of the function f( x )

So from the formula of derivative for polynomial function:

d( x n ) dx =( n x n−1 )

So applying this formula on the given function:

d dx f( x )= d dx ( x n +a x n−1 + a 2 x n−2 +… a n−1 x+ a n ) = d dx ( x n )+ d dx a x n−1 + d dx a 2 x n−2 +…+ d dx a n−1 x+ d dx a n =n x n−1 +a( n−1 ) x n−2 + a 2 ( n−2 ) x n−3 +…+ a n−1 + a n ( 0 ) =n x n−1 +a( n−1 ) x n−2 + a 2 ( n−2 ) x n−3 +…+ a n−1

Since the derivatives of constant is 0

Thus, the derivative of the given function f( x )= x n +a x n−1 + a 2 x n−2 +… a n−1 x+ a n is n x n−1 +a( n−1 ) x n−2 + a 2 ( n−2 ) x n−3 +…+ a n−1 .