For a positive integer

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

m

, let

f_m(x)=x^{m}

for all real

x

. (a) Derive an explicit formula for the

n

th derivative

f_m^{(n)}(x)

for

0\le n\le m

in closed form. Express your result using products of consecutive integers (often called a ``falling factorial''). (b) When

m=10

, determine the smallest integer

n

such that

f_{10}^{(n)}(x)

is identically zero for all

x

. Provide a concise justification based on your result from part (a). (c) Verify your general formula from part (a) by computing

f_{4}^{(4)}(x)

directly through successive differentiation and comparing it to the value predicted by the formula.