Mathematics

Problem

Let

f(x)=x^{5}

. a. (a) Expand

(x+h)^{5}

completely in powers of

h

. b. (b) Simplify

(x+h)^{5}-x^{5}

and then divide the result by

h

. c. (c) Show that the limit of the expression obtained in part (b) as

h\to0

equals

5x^{4}

. Hence, prove that

f'(x)=5x^{4}