Find the derivative of f (x) = (ln(x))2

Please no written by hand

(a) Find the derivative of f (x) = (ln(x))2
(b) Find the partial derivatives of f (x, y) = 5 + 2x3 + √y + 4xαy1−α
(c) Solve the following system of equations for x, y, and z. Show your work.
z + 2y = x
x/y = 5
x · y · z = 15
(1)
(d) A consumer has 10 dollars to spend on apples and bananas. The consumer’s prefer-
ences can be represented with the utility function u(a, b) = a · b, where a is the quantity of
apples and b is the quantity of bananas. Show your work.
If apples cost 1 dollar each, and bananas cost 0.5 dollars each, how many of each fruit
should the consumer buy to make themself as satisfied as possible with their purchase?
Explain why.