Mathematics

Problem

For a positive integer

n

, define

p_n(x)=x^{n}

for all real

x

. (a) Derive a general expression for

p_n'(x)

in terms of

n

and

x

. Your derivation must be valid for all integers

n\ge1

. Clearly state each algebraic step. (b) Using your result from part (a), find

p_6'(2)

. (c) Describe how your formula in part (a) would need to be adapted, if at all, for negative integer exponents.