Find the derivatives of the functions in Exercises 1–42.


𝔂 = (x + 1)² (x² + 2x)

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = cos u, u = −x/3

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

f(x) = { 2x − 1, x ≥ 0

x² + 2x + 7, x < 0

Find the derivatives of the functions in Exercises 1–42.

𝔂 = x⁻¹/² sec (2x)²

Second Derivatives

Find y'' in Exercises 59–64.

y = x(2x + 1)⁴

Power Rule for negative integers Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is,

d/dx (x⁻ᵐ) = −mx⁻ᵐ⁻¹

where m is a positive integer.

The best quantity to order One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is

A(q) = (km / q) + cm + (hq / 2),

where q is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be); k is the cost of placing an order (the same, no matter how often you order); c is the cost of one item (a constant); m is the number of items sold each week (a constant); and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security).

Find dA/dq and d²A/dq².