Show that 1+kx is the local linearization of (1+x) ^k near x = 0

Local linearization L (x) of a function f (x) near x = a is given by f (a) + f' (a) (x - a)

f (x) = (1 + x) ^k

f (0) = (1 + 0) ^k = 1^k = 1

f' (x) = k (1 + x) ^ (k - 1)

f' (0) = k (1 + 0) ^ (k - 1) = k (1) ^ (k - 1) = k (1) = k

Therefore, the local linearization L (x) of (1 + x) ^k near x = 0 is given by f (0) + f' (0) (x - 0) = 1 + k (x) = 1 + kx